278 Prof. Challis's Theoretical Determijiation of 



which is an exact differential of i^f. The further consequences 

 of this supposition, which are remarkable, I proceed to de- 

 velope. 



The above values of u, u, w, give 



dx ""^ dx"' dy~'^ df dz ~*^ dz" ' 

 And since ■^■=.—-c£^fsdt^ it follows that 



Tt-J dt- "-'' 



and 



dt~ a" dt^' 

 Hence, substituting in equation (l.)j 



di^ dz^ ^ f \dx^^ dfJ^ 



Now the nature of the question under consideration requires 

 that this, like the general equation (2.), should be liiiear with 

 constant coefficients. Let therefore the coefficient of <p be 

 equal to a constant —b\ The above equation accordingly 

 resolves itself into the two following : 



H-'^-^'^-' ('•) 



g,.^4^=o. . .... ,., 



which accord precisely with the suppositions already made, 

 that (p is a function of z and t only, andyis a function of x 

 and 7/ only. 



The equation (3.) is transformable into the following, 



^.-|^^ = «' (^•) 



in which iissz + at, and v=z—ai. (See Peacock's Examples, 

 p. 466.) Putting for convenience sake e for -— §, and regard- 

 ing c as a small quantity, the integral of (5.) may be obtained 

 by successive approximations in a series as follows : 



e" 



<p = F{u) + G{v) + e{vF,{u)+uG,{v)}+— x 



