22 M. Falk, On the Integration of partial 



dœ (h/ dz 



whence the general sohition of the given diff. equ. becomes 



I designing a logarithm with the base e and -4/ a third arbitrary function. 

 Ex. 2. The diff. equ. of ruled surfaces: 



'3,0 + 3^-2,1 -{- 3Ä)2ri,2 + <i)^^'o,3 = 0) 



1 — "i,i ih y ~iA — -2,0 -0.2 



where où = — ^ • 



-"O ,2 



The roots of the equation (20) are here all = «; thus (21) and 

 (23) give 



dy — udx = , 



rf'a + 2&j(;L-i 1 -\- cü'^dr^^ = 0, 



where Ave commence by integrating the latter equation. 



This, if multiplied by z^, ± K-ii' — -2,0-0.2 , becomes 



(-1,1 ± iZ-Ll'— -VT^) '^-20 — 2-2 ^^'11 — -2,0 ^^-0,2 = 0- 



Introducing u as a new variable instead of ^i 1 , we have 



w : 



0.2 1 



— -11 dl K "1 .1 ^2 -^o 2 — 



whence ^i ,1 + \/^i 1 ^ — =2 ^0 2 = — -^ ? 



and, by adding the latter to the former, Ave get 



Où 



±2yz.,,-> - - 



-2 -0 2 — ^" -0 2 1 



Où 



and , by subtraction of the latter from the former , 



— 2 s, 1 = uz^,_ + 



Ciù 



This gives by differentiation 



— 2 (7;j 1 := Où d-f) -2 -{- Zo 2 dct) -\- - dz.2 — — -- dcù 



Où &)' 



