Prof. Challis on Hydrodynamics. 265 



to the applications proposed to be eventually made of these 

 researches, and by means of which the relations between the un- 

 known constants may be still further determined. It will be 

 supposed that the radius c of the sphere is so extremely small 

 compared to X, that the distance from the centre at which the 

 sphere has any sensible influence in disturbing the incident 

 waves is very small compared to the same quantity; so that if 



r x represent that distance^ the ratios — and Y are k°th vei T sma ^» 



On this supposition it may be assumed without sensible error 

 that, within the limits of the sphere's influence, the ratio of the 

 condensation at'each point to the condensation of the incident 

 waves is a function of coordinates only. This condition, which, 

 in fact, is involved in the assumed values of/ and F, is equiva- 

 lent to supposing that 



/+F // + £' m . 2iTfcat , .; „ 



— ^—5 \- J = — sin — - — x a function of r. 



r z r tea X 



By substituting the values of/ and F, it will be found that the 

 condition is satisfied if m 2 = — m, and c 2 = c 1 . Hence, eliminating 

 by these equalities m 2 and c 2 , and substituting for/ and F in the 

 foregoing expression for a, the result is 



— = H **— I -5 cos— [r + erf -V -r- sin — - (r + erf ) cos 6. 



<7 l m \r* X * rX X / 



Reverting now to the equation (k) 3 and substituting in it —m l 

 for m 2 , and c x for c 2 , we obtain for determining c x the equation 



2-7T , s X 7TC 



Substituting c for r in the above value of a, and eliminating c Y 

 by means of the last equation, the exact result is 



<r __ m-tfca St^ccosO 



o-, m g/ 4tt 4 c 



X 3 



(.♦*£)• 



which, on account of the small ratio -, is very nearly 



A 



cr wzj 87T 3 /caccos9 



cr-L m X 3 



This equation gives the condensation at any point of the surface 

 of the sphere due to the condensation of the waves, and there- 

 fore additional to that which was before found to be due to the 

 incidence of waves without condensation. It is evident that, 

 since the investigation has been restricted to quantities of the 



