244 Mr. Greatheed's new Method of 



T? i^ dz y dz x , 



Example 2. -=— = ^- —. 1 , 



r dx x dy y 



■n .,. dx y dz x 



By transposition —, -^ — y— = — , 



J r dx x dy y 



d z Id x 



or -j -jr: z = — . 



dx x d log y y 



i d 



-log rr- 



The integrating factor is e diogy ; Multiplying by 



it, and integrating, 



d „ d 



dlogy z = / g ~ 10 S x dlo gy ^ ^ 



logx 



-/£ 



dl °gy d-r 



y 

 a 



y 



^ Q " m "d~T^u 1 . ^/>sy 



dlogy 



Divide by ■ " 0ga?dlo S3/ or * <* lo g?, 



a: 2 - 1 log*-r/— A/ lo S3\ 



therefore 2= -5 — .3/ + « 6 diogy $ (e ). 



2— a 

 diogy 



By the theorem above, 



-1 

 1 -1 y 



: — —y = 3 • 



dlogj/ 



Consequently a = h <p (xy). 



6y 



^ 1 ^ dz , dz 



Example 3. ?/ -^ + •*" -g^- = 2, 



dz ( x d 1 \ 



dx \y dy y ) 



d* (a d A \ 



The integrating factor is 



tJ\ d.rf y * -. g d.yt t& J y % 



or 



