208 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
The terms independent of S' in the right-hand member of this equation add up to zero, 
for x 0 vanishes at (R 2 , S 2 ) ; and, by equating the coefficients of powers of S' in the two 
expressions for x 0 , equations are obtained from which the values of the coefficients B' 
can be found successively. 
33. Relation between Velocity and Time at a Piston. —The time at which any par¬ 
ticular simultaneous values of r and s occur at x 0 = 0, can be found by the method of 
Article 25, and thus the relation between velocity and time at the image of the shot 
may be traced. We can write down the equation 
t- T, = —10- |v" f fer I {(B',-l) + 2BV + 3 B'/+ ... }ds, 
Cl A 2 ^2/ 
in which 
S = {s- S a )/2a, O- = N 2 { 1+ (B' 1 + 1)^ + B'/ + BV 3 +...}, 
and thus t— T 2 can be expanded in powers of S in the form 
t — T 2 = Ci<i + c 2 S 2 + Cg^ 3 + ..., 
where 
Ci = — 
Co = — 
Co = 
c 4 = 
c, = — 
10 h/<r 0 
a 
10 h. 
f CTo 
| 
a 
\S 2 
10h, 
1 (To 
a ( 
\^2 
10 hi 
w 
a ( 
VS 2 ■ 
1—> 
0 
Wo 
a ' 
-—3 
10 
(B'i-1), 
10 
\10 
{B' 3 -V(3B' 1 +l)B' a + 22(B' I +l) !! (B' 1 -l)}, 
,10 
\10 
ifi(B' 1 +l) s (B' 1 -l)}, 
+W (5B', + 3) B' a 2 - W (5B'. ■-1) (», +1 f B'„ 
W(B'. + 1)‘(B'.-1) 
/) 
34. Displacement of a Piston. —To obtain the displacement of the image of the shot, 
we have to find the value of x at x 0 = 0 in terms of simultaneous values of r and s 
occurring on the locus 
r-ll 2 = B\ (s—S a ) + (Bh/Sa) {s-S 2 ) 2 +(B'ftf) {s- S 2 ) 3 + .... 
Now, when x 0 = 0, we have x = ut — Z, and for the value of x at (R>, S 2 ), denoted by 
X 2 , we have X 2 = U 2 T 2 —Z 2 , so that when x 0 = 0, we have 
x-X 2 = ut- U 2 T 2 -(Z-Z 8 ). 
