THE PATH OF A ROTATING SPHERICAL PROJECTILE. 495 



Numerical Approximation to Form of Path. 



17. The differential equations of the trajectory were integrated approximately in 

 § 10 by formally omitting the term in g in the first of them, that is so far as the speed is 

 concerned. In other words : — by assuming that <p is always very small, or the path nearly 

 horizontal throughout. It was pointed out that if the value of <f>, thus obtained from 

 the second, were substituted for sin (f> in the first, equation, we should be able to obtain 

 a second approximation to the intrinsic equation of the path, amply sufficient for all 

 ordinary applications. But the process, though simple enough in all its stages, is long 

 and laborious: — and it is altogether inapplicable to the kinked path, discussed in § 15, 

 which furnishes one of the most singular illustrations of the whole question. 



The fact that one of my Laboratory students, Mr James Wood, had shown himself 

 to be an extremely rapid and accurate calculator led me to attempt an approximate 

 solution of the equations by means of differences : — treating the trajectory as an equi- 

 lateral polygon of 6-foot sides, and calculating numerically the inclination of each to 

 the horizon, as well as the average speed with which it is described. For we may write 

 the differential equations in the form 



1 d(v 2 ) . v 2 



dd> Jc a 



-f = —~COS<b , 



as v v z ^ 



and these involve approximately 



4* -v 2 + 2(- +g sin <p)Ss = , 



Thus we find, after a six-foot step, the new values 



v' 2 = ( 



_^\ 2 _ 384gin 

 a / 



. 6& 192 cos 

 0=0+- -2 



[If we take account of terms in (<^s) 2 , we find that we ought to write for 12/a 

 the more accurate expression 12/a. (1 — 6/a). But this does not alter the form of 

 the expression for v' 2 . It merely increases by some 2 per cent, the denominator of 

 the coefficient of resistance, of which our estimate is, at best, a very rough one ; so 

 that it may be disregarded. But the successive values of v 2 are all on this account 

 too large ; and thus the values of <p, in their turn, are sometimes increased, sometimes 

 diminished, but only by trifling amounts. This is due to the fact that the change of <p 

 depends upon terms having opposite signs ; and involving different powers of v, so that 

 their relative as well as their actual importance is continually changing. These remarks 



