398 Mr. Challis on the general Equations 



and is identical in form with the equation that would be ob- 

 tained from {n), by supposing^ this equation to contain only one 

 of the variables, for instance x : and this plainly should be the 

 case. 



Let r - constant + s, and -^ or -^ = ««. 



dr as 



We have then, 



_d-(}) 2ft) d-(f> _ 1 d^ Po) 



ds' d- — io"'dsdt d- — w-'dt" a^ — w" " 



I will, in the first instance, consider this equation. By treating: 

 it according to the method of Monge, the following two systems 

 of equations will arise : 



ds — {a + w)dt = ] 



, . , , d(p Pwds I ('")• 



(a — w)du)— d.~r- + — — = 

 ^ dt a + w ' 



ds + {a — w)dt = ] 



, d<h Poods \ (s)^ 



(a + co)d(o + d . -P- + = 



' dt a — w ) 



and it is observable that one of these systems is convertible into 

 the other by changing the sign of a. By integrating the equa- 



ds 



tions (»•), putting for dt its value 



a + u) 



pwds 



s — at - I = c, 



»/ a + u) 



u>' dd) , pPwds 



ao) - — — -TT + / = c'. 



2 dt J a + b) 



These two equations are between s, t, c, and c; and c'=/(c); 

 therefore any one of the four quantities is a function of two of 

 them; and as w is a function of s and t, we may put, 



