Prof. Challis on the Principles of Hydrodynamics. 443 



condition (p=i'¥{z—a^t) into the equation (B). Writing for the 

 sake of brevity v for z — a^t, and F for F(v), we shall have 



dfi~''' dv^' dz ~ dv' dzdt~ "^dv^' 

 Consequently, by substituting in (B), 



This equation may be integrated by successive approximations 

 proceeding according to the powers of F. To the first approxi- 

 mation, 



^ + -il-.F=0. 

 dv^ a^ — a^ 



Hence by integrating, 



F = OTCOS {qv + c), 



b b^ 



where q is substituted for — , so that ai^=a^+ -5. Con- 



V a^ — o? q 



sequently 



^-mca%q\z-atsj\+ ^ + ^)- 



By proceeding to the third approximation, I find the following 

 results : — 



^ = m cosq{z—ait + c) 





62 „ „/2o2fl2 



-„«+3+»v(^ + ^). 





Having thus shown that the equation (B) is satisfied by the 

 supposition of a uniform and identical rate of propagation at all 

 points of the axis, and having found approximately the values of 

 ^ and «, to whi'!i this supposition leads, I proceed to consider 

 the integral of that equation in a more general manner. 



It does not appear that an exact integral of (B) can be 

 obtained. An integral, however, applicable to the present 

 inquiry is deducible as follows by successive appi'oximations. 

 For a first approximation, taking the terms of the first order with 

 respect to 0, we have 



5-.'£?+.^*=o w 



