PRINCIPLES OP GUNNERY. 
273 
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zontal velocity n as tlie independent variable, and expressing the four 
quantities <£, t, os, y in terms of u. Thus, dividing (i.) by (ii.), 
du r 
vd(p g 5 
or £ = 
du rv 
■■■jT'-jrz- 
or •-*-#/** .(«) 
Again, from (i.), 
T = f e JSL- ..© 
Jq r COS </> 
A 1 . 4® 
Also, since = «, 
7 7 , udu 
dx — udt —-; 
r cos ^ 
x=rssBJ: . (C ) 
Jq r cos ^ 
Again, since ~ ^ tan <£, 
7 . 7 , u tan<6 du 
du — u tan (h dt =■— 7 : 
* r r cos ({> 
y __ /** « tan^ 
~jq r COS <£ ..... ) 
The above equations— [a), (b), (c),(d )—are the most general forms 
of the integrals which give the characteristic quantities in the motion of 
a projectile. In order to solve them, the form of the quantity f as a 
function of the velocity must be experimentally determined— i.e., the 
law of the resistance of the air to the projectile must be known. 
It has already been shown how this has been accomplished by 
Mr. Bashforth; but as unfortunately the cubic law only holds for a 
limited range of velocity, there is no simple formula for r, so that it is 
necessary to find approximate values of the above integrals. This has 
already been done when the function of the angle is outside the 
integral for the integrals (c), (d) in Table I., and for the integral (b) 
in Table II. (Chap. V., p. 250, &c.) • and Table III. (p. 278) gives, in 
like manner, the approximate value of the integral in equation ( a ). 
In order to make use of these approximations in the solution of 
examples, it is necessary to divide the trajectory into arcs as before, 
and to determine the mean value of the quantity $ over that arc. For 
instance, in the figure at p. 269, in the arc OP , the mean value of <£, 
varying from a to /3, is the inclination of the chord OP to the horizontal 
line. Let this be denoted by <£. 
