solving Equations of partial Differentials. 243 



1 v m 



- . v m *■ — — 



a 4- t», 

 rflog^ 

 where a is any constant. 



Let y = e', then d . log y =z dt, and 



1 1 



,mi 



a ± j~i ^ + -7T 



— a log j/ —a/ 



1 . _ 1 ds mt 1 d?*" - 



771 





+ —5- + 

 a 3 



.nit 



a -£ m a + 77j 



From this it may be shown that if y, y\ y'\ &c. be any 

 number of independent variables, and b 9 b\ b'\ &c. constants, 



_ _ — irw"* 



dlogy dlogy' dlogi/' 



ytn y'm f y\\m" 



a + mb + m f b' + m" b" + 



„ .• dz t dz 

 Example 1. jt-j Hy-^ — = n%. 



d X \ a; d log # x / 



The integrating factor is 



, d , i d 



f 6 dlogy 6 __ x g * dlogy # 



Multiply by this factor, and integrate, therefore, 



x - n .k* "TO * = <p (y) = <> ( 8 l0gy ) 

 d 



and * = x n « " l0g * «^sy $ (i 1<>8y ) 



* , logy -log a? 

 as X $ (s ) 



= x Ha;/ 



2l2 



