274 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



Now, on examination, it appears that the proposed equation is nothing more 

 than the differential coefficient of the quantity — ^ + )n)/-: = added to /; times 



dJ'- 

 the quantity itself : Thus, 



gives 



d^ V dy dm d z d- y „ 



dxi dx •' dx dx dxi 



rf5 y dy d^ y dz (dm \ 



dxi dx d^i ^-^ da; \dx / 



. , , d z -.dm „ 



provided -~-^nz=q and t^ + »««— p=o. 



Thus it appears that the equation is not solved but formed : and this is pro- 

 bably all M. Besge intends. How he can justify his additional remark, that 



—f- + niy = z can be solved if m is a constant, or a linear function of x, I am un- 

 d x^ 



able to conjecture. 



Class 4. Equations 'which we capable of solution by the division of operations. 



17. We have already met with several equations in Class 1, where the total 

 operation was found to be equivalent to the product of two or more partial opera- 

 tions ; and in Art. .9 we have pointed out the manner in which the partial opera- 

 tions are applied, viz., by decomposing the total operation in exactly the same way 

 as an ordinary fraction is decomposed into partial fractions. 



Ex.1. y + axy + bx^+2ax"-p- = 0. 



^ dx* dx 



This equation, when reduced to the symbolical form, is 



y + ae ij + 1 



/-D + 1 ' „ //-D + 2 



.fi ^/3ItDil.4_2a(/4±|-^). A = Oby(D). 

 / — D + i \/ — D + 1 / 



.. /-D + 2 , T. , -D-t-i -D + 1 /-D + l 



Now ' — A = — D + i= =^=-^==== . i 



/-D + 1 ' /-D + i l-D + i /-D + 1 



/-D + 2 ,\ ^ -D + 1 « -D+1 » , ..p.. 



