20 Mr. Herapath on the Integration 



the same as we have given in vol. ii. p. 420 ; for A = — r— r, 

 by the well-known property of equations. 

 Ex. 2. Putting n = 3 our (17) gives 



■$m*»\ c>+ />'-'* { Cl +fe^- ri)x { c+pie-" (19) 



just as we should have it from (6) by putting for A its value 

 — r •— r, — r 2 and for r,, which is aToot of the reduced qua- 

 dratic from the equivalent cubic, r l — r found by (16). The 

 solution however here given, exhibits to advantage the great 

 superiority of our general investigation. Had we been con- 

 tented with the process of successive eduction at first adopted, 

 it would have been almost impossible to develop the simple 

 and elegant law which reigns in the exponents. 



Ex. 3. To carry on the reduction of the general formulae 

 to differentials of the 4th, 5th, &c. orders would be super- 

 fluous, the application being so very obvious. Let us there- 

 fore suppose that any two of the roots in (19) as r t and r 2 are 

 equal. In this case because dx } dx~, are understood as factors 

 respectively to c tn c, (19) is 



(r-r )x 

 r x . tjc , ce v 2 y 



or v = c a e* + c x xe * + — — -r- » 



supposing X = 0. 



Ex. 4. Let us now imagine that all three roots are equal. 

 Our (19) under these circumstances, it is evident gives 



rx . rx . cx * e . rx AY „— rx 



y = c 2 e rx -f c k xe rx -f — - — + e J *&e , 



or when X = 



y = c 2 e rx + c l xe rx + \cx 2 e rx > 



in which as before all three arbitrary constants appear. We 

 have therefore the solutions directly with the full complement 

 of arbitrary constants without any of the refined artifices with 

 which D'Alembert found it necessary to aid the half-guessed 

 imperfect solutions of Lagrange. 



Demonstration of the Completeness of the preceding general 



Solution. 

 Because the roots r, r, , r 2 , r 3 , . . . will admit of a great 

 number of combinations of their differences, besides those we 

 have assumed, it may reasonably be imagined that our general 

 formula (17) or (18) not containing the whole, cannot be com- 

 plete. Our attention will, therefore, now be directed to the 

 consideration of this point. The formula most convenient for 



this 



