272 
PRINCIPLES OP GUNNERY. 
Niven’s 
method of 
solution. 
given tables for the integrals in equations (9), (10), and (11) corres¬ 
ponding to different values of y, between certain ranges of angle. 
Other subsidiary tables are given which materially shorten the cal¬ 
culations. A practical solution of a problem is given at p. 56 of this 
work. 
The value of c } determined by Bashforth, is (vide Chap. V., p. 241) 
d 2 x 
w ' (1000) s ’ 
so that equation (6) becomes 
/loooy = zioooy 
\ «o /w/^7 
whence u 0 (the velocity at the vertex) is determined by taking the 
mean value of K over the arc. 
The value of y is then determined from equation (7) thus : 
Z( «o y 
w * cj \1000/ ’ 
sec 3 
f a sec*0 7 r a tan <£ sec" , /" a sdgj 
The integrals^ "(T3^)* ^ (1^)1 
can then be found from tables at p. 40 et seq., “ Motion of Projectiles,” 
and the values of 1\ X, and Y deduced from equations (9), (10), 
and (11). 
For examples vide p. 60 et seq., “ Motion of Projectiles,” to which 
treatise the reader is referred for this method of solution. 
Another method of solution has been proposed by Mr. W. D. 
Niven, of Trinity College, Cambridge, late Professor of Mathematics to 
the Advanced Class of Poyal Artillery Officers at Woolwich, based 
upon the experiments carried out by Professor Bashforth, and making 
use of his Tables (Chap. V., p. 250, &c.). It has the advantage of 
requiring less tabular matter for the approximate solution of problems 
in practical gunnery. 
Using the same notation as before, and putting r for the retardation 
due to the resistance of the air, the equations of motion obtained by 
resolution in the horizontal direction and in the direction of the normal 
to the trajectory in the ascending branch will be, as before. 
du 
dt 
— r cos </>, 
(i.) 
deb 
= ~9 cos (b .(n.) 
The principle of this method of solution consists in taking the hori- 
