147 
In these eight functions we must express y'in terms of yi and x‘, and then 
eliminate the differentials x’, x’, .... This work of elimination is quite tedious, 
but may be briefly indicated. We construct three functions. 
3Y2¥4 — OY 37 
ay 
Yea 
oie — i (CMEC 
40 
B= 15R, +3R,+-> BR,’ —15 R, R, + 30R, R, 
3y27¥; — ldyeysya + ys? 
- ioienys ae ieee a; (x’)3, 
C=18R, + 3R,—60 R, —21 R, R; — 3? R,* + 35 R,?R,+70R,?R, + 210R,? 
3Y2° Ve —2ly2*¥s¥s + 992 Vs" Ys —*s' Ya" (5, 
— 4 (x ile 
Y2 
and eliminate from these x’, giving the differential invariants 
3 
®,= (8y2*ys —l0 yoysys +42 3°) : (8y2¥4 — dys”) 
@, = (3y2°y6 —21 ya’ Ya¥s + S0Va¥s"¥4 — 3s Ys") : (8y2¥4 — OY37)”. 
MATHEMATICAL DerFinitions. By Mosss C. STEVENS. 
PERFORMANCE OF THE TWENTY-MILLION-GALLON SNow Pumprine ENGINE oF 
THE INDIANAPOLIS WATER Company. By W. F. M. Goss. 
The fact that a pumping engine recently installed within the State of 
Indiana has given a duty performance higher than that previously re- 
ported for any pumping engine in any country is deemed of sufficient 
moment to merit the attention of the Academy. 
This engine was built by the Snow Steam Pump Works of Buffalo, N. Y., 
and its installation at the Riverside station of the Indianapolis Water 
Company was completed in season for an acceptance test in July, 1898. It 
is a triple-expansion, fly-wheel engine, having a single acting pump below 
and in line with each of the three steam cylinders. Its principal dimen- 
sions are as follows: 
Diameter of cylinders: Inches. 
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AN TErMeCIAtE) Fo se. 02 6 Ho Bio Oop cect ceciehaitactn? iC Ina Cacao 52 
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