64 Mr. B. Moon on the Integration of Partial 



-. .j separately = ; therefore 



o = E ?8 +u /^)- T ' (4) 



f(r) + U /'(0 + (5) 



Eliminating f'(s) between (3) and (4), we have 



u£W_s 



H/(t)_ f(t) 



u/'C)~ H /'(0 t ' 



R 3 /'(*) 



u /'(»v 



U/'(r)~ U /'(0~ 



or, putting TvVv = a > and arranging, 



= EV-RTa 2 +USa-U 2 , .... (6) 



an equation the adoption of each of the three roots of which 

 gives us 



0=/'to-«/'O), (7) 



where a is a function of oo and y only. 



The auxiliary equations for the solution of (7) by Lagrange's 

 method are 



= dr + otdt, = ds, = dx, = dy; 



the integration of which gives 



c=r + at, Cx = s, c 2 — a:, c 3 =.y. 



Hence we have 



f(%yrst)=f a {xys(r + at)} =f a (xysfi,) suppose, 



where /j, = r-+-at: and/ a is an arbitrary function. 



Substituting in (3) the value of/ thus derived ; since 



=f a \s), since ~=0, 

 and du 



A*-)=/.V)J "/Wi 



