PRINCIPLES OP GUNNERY. 
275 
Similarly, 
d 2 v . - fp sec0 (1000) 3 ^y 
^ r=Sln Use .* TT-.-W 
It will be seen that the values of X and Y both depend on the 
value of the integral 
given in Table I. (p. 250); so that equations (e) and (/) may be 
written 
d 2 r - 
— X = COS (f) (S q sec <f> — $p sec 0) . (X) 
d 2 . - 
“ I —- sin (ft (S q S ec <j> " $p see 0 ...... • (-$) 
This mode of treatment of the X and Y integrals may be employed 
in the case of the time integral, equation (b) —viz., 
T _ r'p , du ' 
Jq fip) COS 0 ° 
This may be reduced to 
d 2 rr— f v aec $ (1000) 3 dv , v 
W Jq sec* ~~Kv *~~' 3 . W 
or T = T q seG $ — Tp gec 0 . ....( C ) 
the value of which is given in Table II. (p. 252). 
The mean angle <p for the time integral is not, however, quite the 
same as that for the distance integrals, being approximately 
in the ascending branch, and 
a+J3 _ . p_—_q ^ ^ 
T 2 p + q 
in the descending branch. 
It will be observed that the determination of the distance and time 
integrals both depend on the value of the quantity q sec 0, of which 
q is still unknown. This can be found by the consideration of equa¬ 
tion (a). * 
0 fP du rp du 
a ~ p=ff J q wy 
or a-p=fff P -^-r. 
Jq u sec (j)f(u sec 0) 
35 
