274 
PRINCIPLES OP GUNNERY. 
The method of finding the mean value of <£ is too difficult to be 
given in this treatise; but the reader is referred to a paper by 
Mr. Niven “ On the Calculation of the Trajectories of Shot,” in the 
“ Proceedings of the Royal Society,” No. 181, 1877. 
The inclination of the chord OP' is given approximately by 
P — f l 
p + q 
for the ascending branch, and 
v=t±A - 
9 2 
i 
3 P+'1 
(P — a) 
for the descending branch. 
Without going minutely into the whole treatment of the integrals in 
equations (#), (b), ( c ), (d), a general explanation of the method of 
solution will now be given. 
Take, for example, equation (<?), 
p udu 
r cos (j> 
Now, r is some function of v —call it f{v); so that equation (c) 
may be written 
pp udu pi 1 udu , . \ 
= / — - = / - —--- (since v — u sec 0 ). 
J q A v ) cos 0 Ja f\ u sec V cos 0 
X 
If $ is a mean value of the quantity c/> over the given arc requiring 
discussion and determination for this particular integral, 
v PP udu 
then X = / —— — - -k • 
J q f {u sec 0 ) cos (f> 
Suppose now u = v* cos cjE>, then v—u sec <£, and du — cos <j>dv; so 
that 
^ __ P p sec ^ ® cos $ dv 
Jq see 0 f(v ) 
Now, the value of f(v ), determined by Bashforth, 
— iY—Y- 
~ w A VLOOoj’ 
vdv 
cv € 
So that 
_ f*p sec * 
= COS (j) / 
-das 
d 2 
2 sec 0 ^ ^ V 
w \iooo/ 
d* v 
or — X 
U) 
- r 
= COS <f> / 
J a s 
p sec 0 (1000) 3 
q sec 0 
A® 3 
.(«) 
* It is scarcely necessary to warn the reader that the v in this substitution is not the same as 
the v used above for the velocity. It is of course indifferent what symbol may be used in place 
of the new v, as it always occurs inside of a definite integral. 
