350 The Rev. R. Carmichael's Illustrations of 



method, in its full expanded form, 



which may be instantly condensed into the shape 



;sr=y. <£(#*— if). 



In the same manner it may be shown that the integral of the 

 equation 



1 dw 1 dw 1 dw _w 



x dx y dy z dz~ z* 

 is 



w=.z .<£(#*— ?/ 2 , yt—z*, z i —x i ). 



4. The following partial differential equation, which was ori- 

 ginally suggested by Laplace, has been discussed by Sir John 

 Herschel (Transl. of Lacroix, p. 690), 



dz y dz m ■ 

 dx z dy 



Now if we multiply this equation by x } and convert the depend- 

 ent variable, the equation becomes obviously 



Vdx +y dJ ' (n-l)z»-> - X ° l+1 > 

 + u Q {x,y). 



dy/ (n—l)z n 

 and the solution is at once 



1 _ x m+1 



(n— l)s n - ! ~~m + l 



More generally, the integral of the equation 



dw y dw z dw ■ J m n 



t- + - -7- + - -j- — x y zV wq 



dx x dy x dz * 



is 



1 x m + l y n zP 



■ " \q^\)vfl- x ~ S+S+P + 1 + U ° [X ' Vi Z) ' 



5. If it be required to integrate the symmetrical partial differ- 

 ential equation 



dz dz . 



lb' d y = Pq=kz \ . 



we find by the method of Lagrange, as perfected by Charpit, 



where a is an arbitrary constant, and <f> any arbitrary function. 

 Upon inspection of the solution now found, and consideration 

 of the mode in which the partial differential coefficients derived 



