THE PATH OF A ROTATING SPHERICAL PROJECTILE. 435 



already obtained enabled me to interpolate points when (after <£ = 5) those given in the 

 tables were too distant from one another for sure drawing. Another help in completing 

 the curve graphically is given by the fact that the tangent, at any point, makes with the 

 asymptote the angle cp which belongs to the point. This spiral does not, perhaps, exhibit 

 the courses of the two functions so clearly as do the separate curves given by Glaisher ; 

 but it certainly shows their mutual relation, and their maximum and minimum values, in 

 a very striking manner. 



The numbers, affixed to various points of the figured spiral are (in circular measure) the 

 corresponding values of <p, or (by the equations of §5) they may be taken as proportional 

 to the times of reaching these points by the moving ball, starting with infinite speed from 

 an infinite distance. 



8. Even in the plane problem of §5, the introduction of the effects of a steady current 

 of wind in the plane of motion complicates the equations in a formidable manner. Suppose 

 <p be measured from the reversed direction of the wind, and let the speed of the wind be 

 W. Then if U, with direction \^, be the relative velocity of the ball with regard to the 

 wind, (for it is upon this that the resistance, and the deflecting force, depend), we have 



U cos t|t = W -f- s cos <p , 



U sin -»/r = s sin ; 



and the equations of motion are 



. ' U 2 



s= cos(0 — ^) + /cUsin (0 — ^) > 



s 2 U 2 



-= — sin(0-i/r) + MJcos(0-x/r); 



\Xj 



where, once for all, we have written h for Jew. 



Putting v for s, and eliminating t, these become 



dv U 

 v-r- = (Wcos0 + u)+&Wsin$, 



v 2 j = — Wsin0 -f k( W cos + v) ; 



where, of course, 



TJ 2 = W 2 + v* + 2 Wv cos <p. 



These equations reduce themselves at once to the simpler ones above treated, when we 

 put W = 0, and therefore U = v. As they stand they appear intractable, in general, except 

 by laborious processes of quadrature. But while <p is small, i.e., while the ball is advancing 

 nearly in the wind's eye, they may be written approximately as 



dv (W+v) 2 , ,„ T 



From the first of these we see not only that the space-rate of diminution of speed is 

 increased in the ratio (W + vf/v 2 , which was otherwise obvious ; but also that the rotation 

 tends, in a feeble manner, to counteract this effect. From the second we see that the 



VOL. XXXVII. PART II. (NO. 21). 3 U 



