Mr. W. H. L, Russell on the Calculus of Symbols. 433 



equations, which can be put in a form involving these functions exclusively, 

 can be solved like algebraical equations. We shall give some instances of 

 this. 



Consider, first, the equation 

 This may be put in the form 



or 



dy 



we shall have 



Lagrange's method will give in this case the equations 



dy __dx dw^ 



~x~ 1ji~A3o,y)^-{x'+y^)w 

 Hence we shall have 



x'+y'=c\ 



Let rM^^£?)=F(^,c); 

 J Vc— V 



where ^ is an arbitrary function. We shall denote this expression by 

 x(^> y)i whence we have for the determination of (m), 

 du du 



which gives 



(j2+y2)sin-l , ^ , (^2+y^)sin-l — — 



where 



which completes the solution. 

 Next let us take the equation 



od^W O r ^d'^U /n 3 . o 2 , 



