Prof. Challis on Hydrodynamics. 43 



constant c is included, becomes 

 cr=<T l + 2m l . < -gCOs^r + Cj)-!- - sin g (r -f c 2 ) > sin qmt cos 6. 



The determination of the constant c Y having been made on the 

 same principle as before, to determine m x it is only necessary to 

 satisfy the condition that <r = S where r is so large that the 

 sphere's influence is insensible, and consequently the terms in- 

 volving its radius c may be omitted. After expanding sin qr 

 and cos qr, substituting the values of sin qc x and cos qc v and 

 omitting terms of a negligible order, it will be found that 



/2m } q 3 r m,q 3 c 3 \ . , n 



cr=a l — I — ~ 1-— ' a I smq/cat cos v. 



Hence, where r is very large compared to c, 



2m x q 



sin q/cat cos 0. 



Comparing this result with the above value of S, we have 

 Sm sin qc 



Hence where r=c, 



a=a l — m } q 3 c sin qicat cos 6 



Sac sin qc n cos 6 

 = a l -\ — * ip_»i . m sm g##£j 



which gives the condensation at the surface of the sphere so far 

 as it is due to the term in the value of S which involves qr sin qicat. 

 Let us now take account of the term involving 

 q 2 r 2 cos 2 6 sin qicat. 

 The analytical reason for considering this term apart from the 

 other, although it is of a higher order with respect to small 

 quantities, is, that the particular integral in which it is involved 

 satisfies the differential equation (rj) in Part II. independently 

 of the particular integral of the same equation which has just 

 been employed. Our reasoning with respect to this new term 

 will be precisely analogous to that applied to the other. In the 

 first place, we have 



S'= a 1 H — ^r — cos qc sin qtcat cos 2 6, 



and, by the same investigation as in the October Number, 



a' = a x 



+ mA I -3 — |- J cos q (r + c/) + -^ sin q(r + e/) I sin qtcat cos 2 0. 



Determining c/ as before, expanding sin qr and cos qr> and 

 omitting insignificant terms, it will be found that 



,(2q s c b , ? 5 r 2 \ . a 



