216 Prof. Challis on Hydrodynamics. 



it being supposed that-^=0 where p = l. The general inte- 

 gral of the first equation is r(j>=f(r—at)-t-Y(r + at). Now as 

 the value of -~ derived from this integral is the sum of those 

 given by the arbitrary functions taken separately ; and as this is 



true also of -^-, the total motion is the sum of two motions ad- 

 dr 



mitting of independent treatment. Let, therefore, rtf)=f{r—at)j 

 and substitute 1 + a for p. Then, to the first approximation, 

 rf0 = /'(r-afl f(r-at ) aa = f'(r-at) [ 

 dr r r z ' r 



Hence it appears that the condensation a is propagated from the 

 centre with the velocity a, varying at the same time inversely as 

 the distance from the centre. But \ff'(r—at) can be a discon- 

 tinuous function having positive values corresponding to the 

 values of r between r and r + h, and vanishing for all other 

 values of r, and if h, the breadth of the wave of condensation, 

 be constant, the principle of constancy of mass requires that er 

 should vary inversely as the square of the distance. What I have 

 advanced respecting this difficulty in the Philosophical Magazine 

 for January 1859, and in art. 10 in the number for June 1862, 

 was written under the impression that the contradiction might 

 be got rid of by taking account of the second term of the value 



of y~. But I have since discovered that that explanation is 



wholly untenable, and that my original ideas on the question 

 were correct. The reasons for this conclusion are the following. 

 Por the sake of distinctness let the function/' have the specific 



TT 



form m sin j- (r — at + c) , and, a certain time t Q being fixed upon, 

 let the constant c be equal to — r + at . Then by giving to r 



IT 



in the function msin — (r— r ) successive values from r tor + ^ 

 the condensation from point to point of the wave may be obtained. 

 But since /'(r— at) = msin — (r — at + c), it follows that 



fir — at) mh tt , . 



— ~ 2 =— 2Cos r (r — g* + c). 

 r l irr z h ' 



Now it is true that the velocities derived from this expression by 

 substituting for r as above indicated, exactly account for the 

 change of the law of the condensation from the inverse square to 



the simple inverse of the distance, and also that the rate — -%, 



7rr 



