of Linear Differential Equations, 21 



this object is (18), which is precisely the same as (17) but un- 

 der a different form. 



Now that part of (18) which consists of the sums of the ex- 

 ponentials clearly remains the same, whatever commutation 

 takes place in the order of the roots : we must therefore apply 

 our attention to the part affected with the successive integral 

 signs. Taking the lowest integral sign, and integrating by 

 parts, 



/,-»x---r"Jx + 2- + ^- + -£ + .-.. 



of which the general term is 



z+1 



Hence employing the sign <y— to signify produces, gives, &c. 



r + ] 



And because the part under the integral sign is of the same 

 form as before, the general term of the quantity will be 



— r,rr ,2s -^ 



6 «+i *+i- Therefore 

 r i r 



J J ru *+1 r «+I 



Pursuing the same course we at length have 



(r — r )x 



xr r> C r — r ) x /» —rx 



e n ~ l /e n-2n-i ....Je X 



<T 



*+l *+l x+1 

 r . ri ..../ n— 1 



taking no notice of the algebraic sign. This may be regarded 

 as the general term of the last number, and hence as the in- 

 dex of the form it assumes when all the integrations have been 

 executed. But this term w r ould manifestly have the same value 

 and form in whatever order the roots r, r l9 r 2 , . . . be taken; 

 and therefore the member itself would have the same value 

 and form, and consequently the solution we have given com- 

 prehends all the varieties which can arise from the different 

 arrangements that may be given to the roots, and is therefore 

 the complete solution. 



Failure of Lagrange's Method. 



Having established the accuracy and completeness of our 

 own method, we will now transfer our attention to that pro- 

 posed 



