of the Motion of Fluids, ^c. 409 



a., + '^ +^ -/Pdr-2a'f^!^-afPdt = c,+c\=c, + F,ic,) = F(c,). 



Hence 2aa, + af (^ -P)dt- af (^ + P) dt =f{c) + F[c,), 

 2fl= hyp. log. p + aj\ (^ - P) dt + af, (^ + P) rf# =/(c) - F(c,). 



The integral ^1 is to be performed on the supposition that c is 

 constant, and f on the supposition that c, is constant. We 

 cannot hope to effect these integrations; but it is worth while 

 to attend to the case in which i^(c,)=o. In this case 



a>-a hyp. log-, p - f, (~ +P)dt = 0; (H) 



and if r be .so great that the term may be neglected, we 



fall upon the same equation as would be obtained for propaga- 

 tion in a single (the positive) direction, by the consideration 

 of motion in space of one dimension. In fact, the two systems 

 of equations (a) and (/3), which are convertible one into the 

 other by changing the .sign of a, imply that in space of three 

 dimensions, two propagations may exist either simultaneously 

 or separately, one directed towards, the other from, a centre. 



It may be useful to observe that when the motions are small 

 and P = 0, the law of the variation of the velocity very near 

 the centre, is the same as in incompressible fluids. We may 

 confirm this remark in the following manner. Let us suppose 



k 

 in (H) that oi = —, as in the first example of Art. 2. Then, 



/2«^ji „ /■' -kdr , , a-u> 

 — —at=2a 1 J — = — a hyp. log. . 



3G2 



