38 M. Falk, On the Integration of partial 



dp = rdx + sdy and dq = sdx + tdy . . . (70), 



dp -\- udq 

 whence T = r -[- ^ ("*■ +/"•) + tmfji. = 



_ . 'Ss Tt V 



dy 

 since by (65) -y- = m . Tlius the auxiliary system of (68) consists of 



the equations dz = pdx -\- qdy , (70), (69) , (65) and 



R{dp -\- jLidq) = Vdx (71) 



that is: of the system of Monge. 



The equation of Ampere 



Hr 4- 2Ks + Lt -\- M + N {rt — s^) = . . . (72) 



may also be integrated by means of the auxiliary system of (64). 



The eqvation (67) here becomes 



{H + Nt) m' — 2{K — Ns) m -}- L -\- Nr = , 



whence by means of (72) 



K—Ns + VK^ — HL 4- MN 



m = — • 



H-\- m 



Putting G = K' — HL ^ MN (73) 



and using (65), this equation becomes 



Hdy + N{sdx + tdy) — (K ± V'G) dx = 

 or Hdy + Ndq — (K ± \/G) dx = (74) . 



Solving one of the equations between (72) and (73) with respect to 

 - and applying (65), we get 



dx _ K — Ns + VG 



dy ~ L + Nr ' 



whence Ldx -}- NQ-dx + sdy) — (7v + )/G) dy = 



or Ldx + Ndp — (K^iV^dy = (75) 



or, if dy be elimhiated by means of (74) , 



