of Attractive and Repulsive Forces. 183 



rection opposite to that of the incidence and propagation of the 

 waves. Then, if b be the radius of the sphere, the above-men- 

 tioned portion of the transverse plane is equal to 7rb 2 —7rb 2 sin 2 0, 

 or irb 2 cos 2 0. Also, if PM be perpendicular to the axis of the 

 motion, and if OM = £ and 00 = h, we have z = h — b cos 0, and 

 z — not + c = h + c—b cos — Kat. In the subsequent reasoning 



b cos 

 only the first power of the small ratio — will be taken 



into account, and c will be substituted for li + c. 



16. Assuming now that in the incident waves the number 

 of axes which pass through a unit of space is hi, n being a 

 certain standard number and k an arbitrary factor, the number 

 which will pass through the portion irb 2 cos 2 of the transverse 

 plane is krnrb 2 cos 2 0. Hence, if w and S represent respectively 

 the composite velocity and condensation under the supposed 

 conditions of the problem, we have (by the principle stated in 

 art. 10) ^W = hav and $ = kncr. Hence, supposing that 



T=m sin q(c — Kat) ^ — cos 2q(c — Kat), 



m/ Km i . m 2 /2k 6 1 \ - , x 



r= — sm q(c — Kat) 2 I ~q~ + j— 4) cos 2q(c — teat), 



we shall have, to terms inclusive of the first power of — - — , 



A. 



W = 7rknb 2 cos 2 0(T+ -^- . b cos $). 



\ Ka dt / ; 



S = irhib 2 cos 2 (T + -^- b cos + £t\ 

 \ Ka at 4:K*a'/ 



Considering, first, the expression for S, we may omit the terms 



dT 

 containing T' and — as factors, because they are indicative of 

 at 



pressures which are wholly periodic at every point of the 

 sphere's surface, and consequently give rise to no permanent 

 accelerative action. The other part of the value of S is inde- 

 pendent of t y and shows that the sphere is continuously pressed 

 m the direction of the incidence of the waves. The pressure at 

 any point P being « 2 S, the pressure on the whole hemispherical 

 surface estimated in the direction of incidence is 



irhib 2 m 2 



IF 



(Vt^ 2 sin cos 3 6^0, 



taken from 0=- to 0— ^. This gives for the total moving force 



7r 2 knb*m 2 

 Xk* 



