solving Equations of partial Differentials. 245 



The effect of the first part of the factor on the second is to 

 change it into 





x+s/x 2 +y* 



= /*?.-! 



x+y 



■*-i- i 



Multiplying therefore by f rf -3^ -, and integrating, 



By the same principles the equation 



dx dy x+y 

 may be integrated. 



The method is equally applicable to equations of four or 

 more variables, of which I shall take a single example. 



Example 4. 



dz dz dz xy 



u— \- x—j 1- y -7 — = a z H — . 



du dx J dy u 



dz 1 / d d \ xy 



du u \d log x dlogy J u 2 



/ d d \ d d a 



Multiply by e ° gU d lo S * + dlogy ~ ">'_. u dlo S x^d\ogy 9 



and integrate, therefore 



^ d d C d d _ 



g ° g M \dlo~g~x+ d logy ~~ a ) z — C u d log x + d logy ° X y J u 



d d 



dlogz + dlogy" - -1 logx logy 



= W—-j 3 ## + <*>(* ».■ ) 



a a 



c? log # d log 3/ 



—l 



rf log # d log y 



