358 Prof. Challis on the Motion of a small Sphere 



These equations give very approximately 



singc 3 =l, cosgc 3 = — -^r, sin?c' 3 =^p cosgc' 3 =l. 



By substituting in the expression for cr the supposed values of/ 

 and F, and taking account of the relations between the constants, 

 it will be found that 



cr-r a x = 



(jL sin q(r + c 3 ) + (^ ~ |^) cos q(r + c 3 ) j m 3 cos qa't cos 2 6 



-f / % cos q(r + c' 3 ) — f -^ — |r J sin g(r + c' 3 ) j m' 3 cos g«7 cos 2 0. 



After eliminating c 3 and c' 3 by the equations above, expanding 

 the sines and cosines of qr, and neglecting insignificant terms, 

 this equation is reduced to the following : — 



(qb r <z 2q 5 c b \ 

 a~~ + tq.> 3 ) ( m 3 s i n Q a 't + m, 3 cos Q a 't) cos2 ^* 



Then by supposing r to be very much larger than c and neglect- 

 ing the second term in the first brackets, and by equating the 

 result to the term of cr' — a l which contains cos 2 0, we obtain 



w! Q 3 



—} sin q(a't 4- c ) = ^rz (m 3 sin ##'/ -f m' 3 cos ##7), 



_ 45 m' cos qc , _ 45 m' sin qc 

 m3 ~ 2q*a' ' m3 ~ 2^V 



Hence, substituting these values of m 3 and m' 3 in the foregoing 

 equation, and putting c for r, we have for the condensation at 

 any point of the surface of the sphere, 



5m'tf 2 c 2 



/T rr — ■ ± 



6a' 



■sin q(a!t + c ) cos 2 0, 



The velocity along the surface deduced from this value of cr is 



bm'qc . f . n a 



cos q(a't + c ) sm u cos u. 

 o 



Adding now the results of the two integrations, and using a 

 and W to represent the total condensation and velocity at any 

 point of the surface, we have 



Sqcm! . . . „ 5m'o 2 c 2 . , , , x n 

 d — (T l — - t cos q (a't -f c ) cos ~j— sm q (a't + c ) cos 2 d, 



077? itY) (IP 



W= -g- sin g(«'/ -f c ) sin H ^-7- cos q (a't + c ) sin 20. 



