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AMERICAN SOCIETY OF CIVIL ENGINEERS

INSTITUTED 1852


TRANSACTIONS

Paper No. 1172


LOCOMOTIVE PERFORMANCE ON GRADES OF
VARIOUS LENGTHS.

BY BEVERLY S. RANDOLPH, M. AM. SOC. C. E.

WITH DISCUSSION BY MESSRS. C. D. PURDON, JOHN C. TRAUTWINE, JR.,
AND BEVERLY S. RANDOLPH.




In the location of new railways and the improvement of lines already in
operation, it is now well recognized that large economies can be
effected by the careful study of train resistance due to grades and
alignment, distributing this resistance so as to secure a minimum cost
of operation with the means available for construction.

While engaged in such studies some years ago, the attention of the
writer was attracted by the fact that the usual method of calculating
the traction of a locomotive--by assuming from 20 to 25% of the weight
on the drivers--was subject to no small modification in practice.

In order to obtain a working basis, for use in relation to this feature,
he undertook the collection of data from the practical operation of
various roads. Subsequent engagements in an entirely different direction
caused this to be laid aside until the present time. The results are
given in Table 1, from which it will be seen that the percentage of
driver weight utilized in draft is a function of the length as well as
the rate of grade encountered in the practical operation of railways.

In this table, performance will be found expressed as the percentage of
the weight on the drivers which is utilized in draft. This is calculated
on a basis of 6 lb. per ton of train resistance, for dates prior to
1880, this being the amount given by the late A. M. Wellington, M. Am.
Soc. C.. E.,[A] and 4.7 lb. per ton for those of 1908-10, as obtained by
A. C. Dennis, M. Am. Soc. C. E.,[B] assuming this difference to
represent the advance in practice from 1880 to the present time. Most of
the data have been obtained from the "Catalogue of the Baldwin
Locomotive Works" for 1881, to which have been added some later figures
from "Record No. 65" of the same establishment, and also some obtained
by the writer directly from the roads concerned. Being taken thus at
random, the results may be accepted as fairly representative of American
practice.

Attention should be directed to the fact that the performance of the
10-34 E, Consolidation locomotive on the Lehigh Valley Railroad in 1871
is practically equal to that of the latest Mallet compounds on the Great
Northern Railway. In other words, in the ratio between the ability to
produce steam and the weight on the drivers there has been no change in
the last forty years. This would indicate that the figures are not
likely to be changed much as long as steam-driven locomotives are in
use. What will obtain with the introduction of electric traction is
"another story."

These results have also been platted, and are presented in Fig. 1, with
the lengths of grade as abscissas and the percentages of weight utilized
as ordinates. The curve sketched to represent a general average will
show the conditions at a glance. The results may at first sight seem
irregular, but the agreement is really remarkable when the variety of
sources is considered; that in many cases the "reputed" rate of grade is
doubtless given without actual measurement; that the results also
include momentum, the ability to utilize which depends on the conditions
of grade, alignment, and operating practice which obtain about the foot
of each grade; and that the same amount of energy due to momentum will
carry a train farther on a light grade than on a heavy one.

There are four items in Table 1 which vary materially from the general
consensus. For Item 9, the authorities of the road particularly state
that their loads are light, because, owing to the congested condition of
their business, their trains must make fast time. Item 10 represents
very old practice, certainly prior to 1882, and is "second-hand." The
load consisted of empty coal cars, and the line was very tortuous, so
that it is quite probable that the resistance assumed in the calculation
is far below the actual. Items 15 and 17 are both high. To account for
this, it is to be noted that this road has been recently completed,
regardless of cost in the matter of both track and rolling stock, and
doubtless represents the highest development of railroad practice. Its
rolling stock is all new, and is probably in better condition to offer
low resistance than it will ever be again, and there were no "foreign"
cars in the trains considered. The train resistance, therefore, may be
naturally assumed to be much less than that of roads hauling all classes
of cars, many of which are barely good enough to pass inspection. As the
grades are light in both cases, this feature of train resistance is
larger than in items including heavier grades. Attention should be
called to the fact that a line connecting the two points representing
these items on Fig. 1 would make only a small angle with the sketched
curve, and would be practically parallel to a similar line connecting
the points represented by Items 13 and 16. There is, therefore, an
agreement of ratios, which is all that needs consideration in this
discussion.

[Illustration: FIG. 1.--DIAGRAM SHOWING PERCENTAGE OF WEIGHT ON DRIVERS
WHICH IS UTILIZED IN TRACTION ON GRADES OF VARIOUS LENGTHS]

Wellington, in his monumental work on railway location, presents a table
of this character. The percentages of weight on the drivers which is
utilized in draft show the greatest irregularity. He does not give the
length of the grades considered, so that it is impossible to say how far
the introduction of this feature would have contributed to bring order
out of the chaos. In his discussion of the table he admits the
unsatisfactory character of the results, and finally decides on 25% as a
rough average, "very approximately the safe operating load in regular
service." He further states that a number of results, which he omits for
want of space, exceeds 33 per cent. The highest shown in Table 1 will be
found in Item 1 (0.06 mile, 0.066 grade), showing 33 per cent. There is
no momentum effect here, as the grade is a short incline extending down
to the river, and the start is necessarily a "dead" one. The reports of
Item 3, which shows 31%, and Item 5, which shows 27%, state specifically
that the locomotives will stop and start the loads given at any point on
the grade.

The results of a series of experiments reported by Mr. A. C. Dennis in
his paper, "Virtual Grades for Freight Trains," previously referred to,
indicate a utilization of somewhat more than 23%, decreasing with the
speed.

All this indicates that the general failure of locomotives to utilize
more than from 16 to 18% on long grades, as shown by Table 1, can only
be due to the failure of the boilers to supply the necessary steam.
While the higher percentage shown for the shorter grades may be ascribed
largely to momentum present when the foot of the grade is reached, the
energy due to stored heat is responsible for a large portion of it.

When a locomotive has been standing still, or running with the steam
consumption materially below the production, the pressure accumulates
until it reaches the point at which the safety valve is "set." This
means that the entire machine is heated to a temperature sufficient to
maintain this pressure in the boiler. When the steam consumption begins
to exceed the production, this temperature is reduced to a point where
the consumption and production balance.

The heat represented by this difference in temperature has passed into
the steam used, thus adding to the energy supplied by the combustion
going on in the furnace. The engines, therefore, are able to do
considerably more work during the time the pressure is falling than they
can do after the fall has ceased.

The curve in Fig. 1 would indicate that the energy derived from the two
sources just discussed is practically dissipated at 15 miles, though the
position of the points representing Items 16, 18, 19, 20, and 21 would
indicate that this takes place more frequently between 10 and 12 miles.
From this point onward the performance depends on the efficiency of the
steam production, which does not appear to be able to utilize more than
16% of the weight on the drivers. The diagrams presented by Mr. Dennis
in his paper on virtual grades, and by John A. Fulton, M. Am. Soc.
C. E., in his discussion of that paper, indicate that similar results
would be shown were they extended to include the distance named.

From this it would appear that a locomotive is capable of hauling a
larger train on grades less than 10 miles in length than on longer
grades, and that, even when unexpectedly stopped, it is capable of
starting again as soon as the steam pressure is sufficiently built up.
Conversely, it should be practicable to use a higher rate of ascent on
shorter grades on any given line without decreasing the load which can
be hauled over it. In other words, what is known as the "ruling grade"
is a function, strictly speaking, of the length as well as the rate of
grade.

In any discussions of the practicability of using a higher rate on the
short grades, which the writer has seen, the most valid objection has
appeared to be the danger of stalling and consequent delay. As far as
momentum is relied on, this objection is valid. Within the limits of the
load which can be handled by the steam, it has small value, as it is
only a question of waiting a few minutes until the pressure can be built
up to the point at which the load can be handled. As this need only be
an occasional occurrence, it is not to be balanced against any material
saving in cost of construction.

The writer does not know of any experiments which will throw much light
on the value of heat storage as separated from momentum, though the
following discussion may prove suggestive:

A train moving at a rate of 60 ft. per sec., and reaching the foot of a
grade, will have acquired a "velocity head" of 56.7 ft., equivalent to
stored energy of 56.7 × 2,000 = 113,400 ft-lb. per ton. On a 0.002
grade (as in Item 15 of Table 1) the resistance would be, gravity
4 lb. + train 4.7 lb. = 8.7 lb., against which the energy above given
would carry the train through 113,400 ÷ 8.7 = 13,034 ft., say, 2.5
miles, leaving 5 miles to be provided for by the steam production.
Examining the items in the table having grades in excess of 10 miles, it
will be noted that 16% is about all the weight on drivers which can be
utilized by the current supply of steam. In Item 15 the energy derived
from all sources is equivalent to 24.3%; hence the stored heat may be
considered as responsible for an equivalent of 24.3% - 16% = 8.3% for a
distance of 5 miles.


TABLE 1.

===========================================================================
Item No.
  |Length of grade, in miles.
  |     |Rate of grade.
  |     |      |Maximum curvature.
  |     |      |      |Compensation.
  |     |      |      |    |Gross weight of load, in tons.
  |     |      |      |    |     |Weight of tender, in tons.
  |     |      |      |    |     |  |Weight of locomotive, in tons.
  |     |      |      |    |     |  |     |Weight on drivers, in tons.
  |     |      |      |    |     |  |     |     |Percentage of weight on
  |     |      |      |    |     |  |     |     |drivers utilized in draft.
  |     |      |      |    |     |  |     |     |     |Class.
  |     |      |      |    |     |  |     |     |     |
  |     |      |      |    |     |  |     |     |     |
--+-----+------+------+----+-----+--+-----+-----+-----+--------------------
 1| 0.06|0.066 |      |    |  115|  | 37.5| 29  |0.358| 8-28-1/3 C
 2| 0.33|0.0203|25°20'|    |  242|25| 35  | 23  |0.285| 8-28     C
 3| 1.0 |0.06  |16°   |0.05|  192|22| 57.5| 50  |0.310|10-36     E
 4| 1.3 |0.0127|      |    |  600|16| 40  | 32.5|0.300|Mogul.
 5| 1.4 |0.0128| 3°12'|    |  750|15| 51  | 44  |0.270|10-34     E
 6| 2.0 |0.01  |      |    |1,000|15| 51  | 44  |0.291|10-34     E
 7| 2.2 |0.013 | 3°   |    |  725|15| 51  | 44  |0.245|10-34     E
 8| 2.5 |0.0144| 6°   |    |  400|27| 42  | 32  |0.237|10-32     E
 9| 2.5 |0.004 |      |    |2,700|70| 96.7| 85.8|0.207| H 6 - A
10| 3.5 |0.033 |14°   |    |  100|25| 35  | 35  |0.160|
11| 3.6 |0.035 |10°   |0.05|  236|22| 57.5| 50  |0.245|10-36     E
12| 4.0 |0.0085| 4°   |    |1,020|30| 51  | 44  |0.256|10-34     E
13| 6.0 |0.0145|      |    |  308|25| 38  | 28  |0.207|10-28     D
14| 6.0 |0.020 |10°   |0.05|  460|32| 57.5| 50  |0.242|10-34     E
15| 7.5 |0.002 |      |  C |6,152|86|134.5|109.5|0.243|Mallet.
16| 9.75|0.018 |      |    |  200|18| 29  | 29  |0.170|
17|10.0 |0.006 |      |  C |6,173|86|299  |265  |0.203|Mallet.
18|12.0 |0.018 |10°   |    |  280|30| 51  | 44  |0.160|10-34     E
19|12.0 |0.022 |      |    |  850|74|175  |156  |0.166|D-D 16
20|13.0 |0.022 |      |    |  800|74|177  |158  |0.153|D-D  1
21|13.0 |0.022 |14°   |    |  415|50| 91  | 83  |0.154|Consol.
22|16.0 |0.0044|      |    |  950|30| 51  | 44  |0.164|10-34     E
23|20.0 |0.022 |      |    |  500|62| 97.5| 90  |0.170|F 8, Consol.
24|20.0 |0.022 |      |    |  800|74|177  |158  |0.159|L-1, Mallet.
===========================================================================


============================================================================
  |Maker.  |Railroad.                   |Reporting Officer.           |Year.
--+--------+----------------------------+-----------------------------+-----
 1|Baldwin.|Morgan's Louisiana & Texas  |Newell Tilton, Asst. Supt.   |1880
 2|   "    |Long Island                 |S. Spencer, Gen. Supt.       |1878
 3|   "    |Atchison, Topeka & Santa Fe |J. D. Burr,  Asst. Engr.     |1879
 4|   "    |Chillan & Talcahuana        |J. E. Martin, Local Supt.    |1879
 5|   "    |Chicago, Burlington & Quincy|H. B. Stone                  |1880
 6|   "    |Chicago, Burlington & Quincy|      "                      |1880
 7|   "    |Chicago, Burlington & Quincy|      "                      |1880
 8|   "    |St. Louis & San Francisco   |C. W. Rogers, Gen. Mgr.      |1879
 9|Pa. R.R |Cumberland Valley.          |                             |1910
10|        |                            |                             |1910
11|Baldwin.|Atchison, Topeka & Santa Fe |J. D. Burr,  Asst. Engr.     |1879
12|   "    |Missouri Pacific            |John Hewitt, Supt. M. P.     |1880
13|   "    |Western Maryland            |D. Holtz, M. of Mach'y.      |1878
14|   "    |Atchison, Topeka & Santa Fe |J. D. Burr,  Asst. Engr.     |1879
15|   "    |Virginian Ry.               |                             |1910
16|        |Pennsylvania                |                             |1910
17|Baldwin.|Virginian Ry.               |                             |1910
18|   "    |Lehigh Valley, Wyoming Div. |A. Mitchell, Div. Supt.      |1871
19|   "    |Great Northern              |Grafton Greenough.           |1908
20|   "    |Great Northern              |Grafton Greenough.           |1908
21|   "    |Baltimore & Ohio            |F. E. Blaser, Div. Supt.     |1910
22|   "    |Central of N. J.            |W. W. Stearns, Asst.Gen.Supt.|1880
23|   "    |Great Northern              |Grafton Greenough.           |1908
24|   "    |Great Northern              |Grafton Greenough.           |1906
============================================================================


==============================================================================
  |Source of Data.                          |Remarks.
--+-----------------------------------------+---------------------------------
 1|Baldwin Catalogue, 1881, p. 134          |
 2|   "        "      1881, "   72          |10 miles per hour.
 3|   "        "      1881, "  115          | 8   "    "   "
  |                                         |    Stops and starts on grade.
 4|   "        "      1881, "  100          |
 5|   "        "      1881, "  116          |Stops and starts at any point
  |                                         |    on grade.
 6|   "        "      1881, "  116          |
 7|   "        "      1881, "  116          |
 8|   "        "      1881, "   87          |
 9|                                         |
10|Trautwine's Pocket Book, Ed. 1882, p. 412|Empty cars; many curves and
  |                                         |    reversions.
11|Baldwin Catalogue, 1881, p. 114          |
12|   "        "      1881, "  112          |
13|   "        "      1881, "   86          |12 miles per hour.
14|   "        "      1881, "  114          | 8   "    "   "
15|_Engineering News_, Jan. 13, 1910.       |
16|Trautwine's Pocket Book, Ed. 1882, p. 412|
17|_Engineering News_, Jan. 13, 1910.       |Road locomotive and helper.
18|Baldwin Catalogue, 1881, p. 112          |
19|Baldwin Loco. Wks. Record, No. 65, p.  29|
20|Baldwin Loco. Wks. Record, No. 65, p.  29|
21|                                         |Very crooked line. Uncompensated.
22|Baldwin Catalogue, 1881, p. 113          |
23|Baldwin Loco. Wks. Record, No. 65, p.  29|
24|Baldwin Loco. Wks. Record, No. 65, p.  29|
==============================================================================

In proportioning grade resistance for any line, therefore, a locomotive
may be counted on to utilize 24.3% of the weight on the drivers for a
distance of 5 miles on a 0.002 grade without any assistance from
momentum, and, in the event of an unexpected stop, should be able, as
soon as a full head of steam is built up, to start the train and carry
it over the grade. This is probably a maximum, considering the condition
of the equipment of this Virginian Railway, as previously mentioned.

Treating Item 14 in the same way, a distance of 2,310 ft. is accounted
for by momentum, leaving, say, 5.5 miles for the steam, or the length of
a 0.02 grade on which a locomotive may be loaded on a basis of tractive
power equal to 24.2% of the weight on the drivers.

From these figures it may be concluded that on lines having grades from
12 to 15 or more miles in length, grades of 3 to 5 miles in length may
be inserted having rates 50% in excess of that of the long grades,
without decreasing the capacity of the line. This statement, of course,
is general in its bearings, each case being subject to its especial
limitations, and subject to detailed calculations.

It may be noted that the velocity of 60 ft. per sec., assumed at the
foot of the grade, is probably higher than should be expected in
practice; it insures, on the other hand, that quite enough has been
allowed for momentum, and that the results are conservative.

Arguments like the foregoing are always more or less treacherous; being
based on statistics, they are naturally subject to material
modifications in the presence of a larger array of data, therefore,
material assistance in reaching practical conclusions can be given by
the presentation of additional data.




DISCUSSION


C. D. PURDON, M. AM. SOC. C. E. (by letter).--Some years ago the writer,
in making studies for grade revision, found that the tractive power of a
locomotive up grade becomes less as the length of the grade increases,
and in some unknown proportion. This was a practical confirmation of the
saying of locomotive engineers, that the engine "got tired" on long
grades. On a well-known Western railroad, with which the writer is
familiar, experiments were made for the purpose of rating its
locomotives. The locomotives were first divided into classes according
to their tractive power, this being calculated by the usual rule, with
factors of size of cylinders, boiler pressure, and diameter of drivers,
also by taking one-fourth of the weight on the drivers, and using the
lesser of the two results as the tractive power.

Locomotives of different classes, and hauling known loads, were run over
a freight division, the cars being weighed for the purpose; thus the
maximum load which could be handled over a division, or different parts
of a division, was ascertained, and this proportion of tonnage to
tractive power was used in rating all classes.

Of course, this method was not mathematically accurate, as the condition
of track, the weather, and the personal equation of the locomotive
engineers all had an effect, but, later, when correcting the rating by
tests with dynamometers, it was found that the results were fairly
practical.

There were three hills where the rate of grade was the same as the rest
of the division, but where the length was much in excess of other grades
of the same rate.

Designating these hills as _A_, _B_, and _C_, the lengths are,
respectively, 2.44, 3.57, and 4.41 miles. There were no other grades of
the same rate exceeding 1 mile.

In one class of freight engines, 10-wheel Brooks, the weight of the
engine was 197,900 lb.; tender, 132,800 lb.; weight on drivers, 142,600
lb.; boiler pressure, 200 lb.; and tractive power of cylinders, 33,300
lb.

On Hill _A_ these engines are rated at 865 tons, as compared with 945 on
other parts of the division. As the engine weighs 165 tons and the
caboose 15 tons, 180 tons should be added, making the figures, 1,045 and
1,125 tons. Thus the length of the grade, 2.44 miles, makes the tractive
power on it 92% of that on shorter grades.

On Hill _B_, the rating, adding 180 tons as above, is 1,160 and 1,230
tons, respectively, giving 94% for 3.57 miles.

On Hill _C_, the rating, with 180 tons added, is 1,130 and 1,230 tons,
making 92% for 4.41 miles.

Taking the same basis as the author, namely, 4.7 lb. per ton, rate of
grade × 20, and weight on drivers, gives:

Hill _A_,  18.078%, remainder of division, 19.462%
Hill _B_,  20.068%,     "     "      "     21.279%
Hill _C_,  19.549%,     "     "      "     21.279%

It will be noted that the author uses the weight on the drivers as the
criterion, but the tractive power is not directly as the weight on the
drivers, some engines being over-cylindered, or under-cylindered; in the
class of engines above mentioned the tractive power is 23.35% of the
weight on the drivers.

The writer made a study of several dynamometer tests on Hill _C_. There
is a grade of the same rate, about 1 mile long, near this hill, and a
station near its foot, but there is sufficient level grade between this
station and the foot of the hill to get a good start.

All the engines of the above class, loaded for Hill _C_, gained speed on
the 1-mile grade, but began to fall below the theoretical speed at a
point about 2-1/4 miles from the foot of the hill. This condition
occurred when the trains stopped at the station and also when they
passed it at a rate of some 16 or 18 miles per hour, the speed becoming
less and less as the top of the hill was approached.

The writer concludes that the author might stretch his opinion as to
using heavier rates of grade on shorter hills than 10 miles, and indeed
his diagram seems to intimate as much, and that, for economical
operation, the maximum rate of grade should be reduced after a length of
about 2 miles has been reached, and more and more in proportion to the
length of the hill, in order that the same rating could be applied all
over a division.

This conclusion might be modified by local conditions, such as an
important town where cars might be added to or taken from the train.

While it does not seem practicable to the writer to calculate what the
reduction of rate of grade should be, a consensus of results of
operation on different lengths of grade might give sufficient data to
reach some conclusion on the matter.

The American Railway Engineering and Maintenance of Way Association has
a Committee on "Railway Economics," which is studying such matters, but
so far as the writer knows it has not given this question any
consideration.

The writer hopes that the author will follow up this subject, and that
other members will join, as a full discussion will no doubt bring some
results on a question which seems to be highly important.


JOHN C. TRAUTWINE, JR., ASSOC. AM. SOC. C. E. (by letter).--In his
collection of data, Mr. Randolph includes two ancient cases taken from
the earliest editions (1872-1883) of Trautwine's "Civil Engineer's
Pocket-Book," referring to performances on the Mahanoy and Broad
Mountain Railroad (now the Frackville Branch of the Reading) and on the
Pennsylvania Railroad, respectively.

In the private notes of John C. Trautwine, Sr., these two cases are
recorded as follows:

     "On the Mahanoy & Broad Mtn. R. R., _tank_ Engines of 35 tons, _all
     on 8 drivers_, draw 40 _empty_ coal cars weighing 100 tons, _up_ a
     continuous grade of 175 ft. per mile for 3-1/2 miles; & around
     curves of 450, 500, 600 ft. &c. rad., at 8 miles an hour. (1864)
     This is equal to 77-14/100 tons for a 27-ton engine." (Vol. III, p.
     176.)

     "On the Penn Central 95 ft. grades for 9-3/4 miles, a 29-ton engine
     all on 8 drivers takes 125 tons of freight and 112 tons of engine,
     tender, & cars, in all 237 tons,[C] and a passenger engine takes up
     3 cars at 24 miles an hour (large 8 wheels). When more than 3, an
     auxiliary engine."

It will be seen that Mr. Randolph is well within bounds in ascribing to
the Mahanoy and Broad Mountain case (his No. 10) a date "certainly prior
to 1882," the date being given, in the notes, as 1864; while another
entry just below it, for the Pennsylvania Railroad case, is dated 1860.

It also seems, as stated by Mr. Randolph, quite probable that the
frictional resistance (6 lb. per 2,000 lb.) assumed by him in the
calculation is far below the actual for this Case 10. The small, empty,
four-wheel cars weighed only 4,400 lb. each. Furthermore, the "tons," in
the Trautwine reports of these experiments, were tons of 2,240 lb. On
the other hand, the maximum curvature was 12° 45' (not 14°, as given by
the author), and the engine was a tank locomotive, whereas the author
has credited it with a 25-ton tender.

After making all corrections, it will be found that, in order to bring
the point, for this Case 10, up to the author's curve, instead of his 6
lb. per 2,000 lb., a frictional resistance of 66 lb. per 2,000 lb. would
be required, a resistance just equal to the gravity resistance on the
3.3% grade, making a total resistance of 132 lb. per 2,000 lb.

While this 66 lb. per ton is very high, it is perhaps not too high for
the known conditions, as above described. For modern rolling stock, Mr.
A. K. Shurtleff gives the formula:[D]

Frictional resistance, on tangent, }
in pounds per 2,000 pounds         } = 1 + 90 ÷ C,

where _C_ = weight of car and load, in tons of 2,000 lb. This would
give, for 4,400-lb. (2.2-ton) cars, a frictional resistance of 42 lb.
per 2,000 lb.; and, on the usual assumption of 0.8 lb. per 2,000 lb. for
each degree of curvature, the 12.75° curves of this line would give 10
lb. per ton additional, making a total of 52 lb. per 2,000 lb. over and
above grade resistance, under modern conditions.

In the 9th to 17th editions of Trautwine (1885-1900), these early
accounts were superseded by numerous later instances, including some of
those quoted by the author.

In the 18th and 19th editions (1902-1909) are given data respecting
performances on the Catawissa Branch of the Reading (Shamokin Division)
in 1898-1901. These give the maximum and minimum loads hauled up a
nearly continuous grade of 31.47 ft. per mile (0.59%) from Catawissa to
Lofty (34.03 miles) by engines of different classes, with different
helpers and without helpers.

Table 2 (in which the writer follows the author in assuming frictional
resistance at 4.7 lb. per 2,000 lb.) shows the cases giving the maximum
and minimum values of the quantity represented by the ordinates in the
author's diagram, namely, "Traction, in percentage of weight on
drivers."

It will be seen that the maximum percentage (16.1) is practically
identical with that found by the author (16) for grade lengths exceeding
17 miles.

Near the middle of the 34-mile distance there is a stretch of 1.51
miles, on which the average grade is only 5.93 ft. per mile (0.112%),
and this stretch divides the remaining distance into two practically
continuous grades, 19.39 and 13.13 miles long, respectively; but, as the
same loads are hauled over these two portions by the same engines, the
results are virtually identical, the maxima furnishing two more points
closely coinciding with the author's diagram.


TABLE 2.--TRACTIVE FORCE, CATAWISSA TO LOFTY.

========================================================================
Length of grade, in miles                           |    |    34.03
                                                    |    |
Grade {in feet per mile                             |    |    31.47
      {percentage                                   |_A_ |     0.597
                                                    |    |
Resistances, in pounds per 2,000 lb.,               |    |
    Gravity (=20 _A_) = 11.94. Friction = 4.70      |_B_ |    16.64
                                                    |    |
  Load:      | Cars.  | Locomotive.| Tender.        |    |
  Maximum[E] | 1,561  |   44.60    |  25.25         |_C_ | 1,631
  Minimum[F] | 1,031  |   60.50    |  34.50         |_C_ | 1,126
                                                    |    |
Traction (= _B_ _C_ ÷ 2,000 ) Maximum[E]            |_D_ |    13.60
                              Minimum[F]            |_D_ |     9.38
 Weight on Drivers:   | Locomotive.|  Helper.       |    |
    Maximum[E]        |    21.60   |    63.00       |_E_ |    84.60
    Minimum[F]        |    47.00   |    72.00       |_E_ |   119.00
                                                    |    |
Percentage ( = _D_ ÷ _E_ ).                         |    |
  Maximum                                           |_F_ |    16.1
  Minimum                                           |_F_ |     7.9
========================================================================

    FOOTNOTES:

    [Footnote E: Giving maximum values of percentage, _F_.]

    [Footnote F: Giving minimum values of percentage, _F_.]



BEVERLY S. RANDOLPH, M. AM. SOC. C. E. (by letter).--The percentages
given by Mr. Purdon would seem to indicate that the length of the grades
did not affect the loads in the cases cited, but these percentages are
so much below those shown in the table, for similar distances, as to
indicate some special conditions which the writer has been unable to
find in the text.

The use of the percentage of weight on drivers which is utilized in
traction as a measure of the efficiency of the locomotive, while,
probably, not applicable to individual machines, is sound for the
purposes of comparison of results to be obtained on various portions of
a line as far as affected by conditions of grade and alignment. It has
the advantage of disregarding questions of temperature, condition of
track, character of fuel, etc., which, being the same on all portions of
the line, naturally balance and do not affect the comparison. It is, of
course, simply a method of expressing the final efficiency of the
various parts of the locomotive, and, since it depends entirely on
actual results already accomplished, leaves no room for difference of
opinion or theoretical error.

The writer has always considered an "under-cylindered" locomotive as a
defective machine. All weight is a distinct debit, in the shape of wear
and tear of track and running gear, resistance due to gravity on grades,
interest on cost, etc. When this weight fails to earn a credit in the
way of tractive efficiency, it should not be present.

The statement relative to the performance of locomotives on "Hill _C_"
is interesting, especially in that it appears to have been immaterial
whether they made a dead start after stopping at the station or
approached the foot of the hill at 16 to 18 miles per hour. The momentum
would appear to be an insignificant factor.

It is gratifying to note that Mr. Trautwine has been able to brace up
the weak member of Table 1 so completely with his detailed data; also
that his other results strengthen the conclusions reached in the
paper.


    FOOTNOTES:

    [Footnote A: "The Economic Theory of Railway Location," 1887 edition,
    p. 502.]

    [Footnote B: _Transactions_, Am. Soc. C. E., Vol. L, p. 1.]

    [Footnote C: "Nearly 200 tons _exclusive_ of eng. & ten." (Vol. III,
    p. 176-1/10.)]

    [Footnote D: American Railway Engineering and Maintenance of Way
    Association, Bulletin 84, February, 1907, p. 99.]