of the Motion of Fluids, Src. 40f 



w* dd) J, J , 

 ao,----f^+gf<^dt=^c. 



Hence a<«-^ - -^ + g{s-at) = c' + gc=f,{,c)+gc=f{c). 

 So from the equations (s), 



a<" + ^ + -^-^(s + a<) = c', =gc, = F,(c,)-g-c, = F{c,). 



From these two equations, by reason of the equation 



«" hyp. log. /> = gs - ^- y, 



we obtain, 



2a{u,-gl)=f{c)^F{c,). 

 2a' hyp. log. p =/{c) - F(c.). 



Of these functions, / refers to propaaration in the positive di- 

 rection, F to propagation in the negative. To obtain a particular 

 integral, suppose that F(c,) = o. Then 



a,-gt = a hyp. log. p =-^-^ 

 Hence /.dt =4 +'^.t = u,t-i^; 



'' 2 2fl 2 



2 



or, becau.se dw = gdt, fwdt = ^ . Hence finally, 

 w- gt = a hyp. log. p=f(^s-at-wt + ^^; 



or, a, - g« = a hyp. log. ^ =/(s - a< - ^) . 



Both these integrals will be found to satisfy the given equations. 

 If in the first we make s and i vary so that p does not 

 alter, we find ds — adt - wdt — tdw + gtdt = o. But as p does 



Vol. III. Part III. 3G 



