44 Prof. Challis on Hydrodynamics. 



Hence where r is large we have 



S^o*! -A™ — sin qicat cos 2 0. 



Comparing this with the above value of S', it follows that 



n cc 

 27/ 



, 4omcos(7c 

 m,'= 



Then putting c for r, 



mfq b c 



sin q/cat cos 2 



= a ] H ^ — cos </c sin qicat cos 2 9, 



which gives the condensation at the surface of the sphere due to 

 the term involving q-?- 2 cos 2 sin q/cat. 



It will be remarked that although the constants m x and ?w/ have 

 thus been determined, a new constant c has been introduced, 

 the character and value of which remain to be ascertained in 

 order to complete the solution of the problem. This I hope to 

 be able to do on a future occasion. At present I will only re- 

 mark that if qc be very small we may put in the small terms 

 Ka<r x for m sin quit. In that case the expressions for a and a' will 

 both have the form o-^l+mQ), and, by the same reasoning as 

 that employed in Part II., it may be shown that to take into 

 account the omitted quantity Acr l it suffices to substitute 

 <Tj -|- Act, for a l in the two formulae. The tendency of the action 

 of the waves to produce a permanent motion of translation of 

 the sphere may then be deduced from the function Aa l as before ; 

 and on the supposition that the waves emanate from a centre, 

 the law of the inverse square is satisfactorily referable to the 

 composition of this quantity, inasmuch as the number of 

 axes of motion passing through a given area, to which the 

 number of terms it involves has been shown to correspond, evi- 

 dently varies inversely as the square of the distance. As my 

 hydrodynamical researches have led me to conclude that the 

 velocity (V) in central motion varies inversely as the square of 



V 2 

 the distance, the quantity — , to which Aa l in the former inves- 



KG- 



tigation was supposed to be equal, would for such motion vary 

 inversely as the fourth power of the distance. This particular 

 being excepted, I have seen no reason either to correct, or to add 

 to, the remarks made in Part II. relative to the bearing of these 

 Researches on a mathematical theory of attractive and repulsive 

 forces. 



I have further to state that, having gone through investiga- 

 tions analogous to the two preceding, relative to the two terms 



