118 



Iroduce into <p.x an arbitrary term independent of x. We have 

 next to consider the case of two roots being equal to zero ; but 

 as this is included in tlie more general ca^e of two roots being 

 equal to each other, I shall proceed to this last ; and suppose that 



the equation, V = o, has two roots represented by « + h$ a and 

 a+ h' $ «, which are of course ultimately equal to each other. 

 The terms of (p.x corresponding to these roots are Be ^'"^ *^ " 



and B c ; which being developed according to the powers 



of the indefinitely small variation 5 «, and added together give 



{B + B)c''' + (Bh + Bh')c''^x'U+{Bh--+Bh'0 «""^-^7^ + 

 &c. (1) Instead of the arbitrary constants B and B, we may feign 

 two others A and A', so that B + B = A, and Bh+ Bh'= A'. 



In fact, this condition will be fulfilled, if we make 



B = -^~^''' and B = '^^~^. Substituting these values for 



ir^^i h -Hi 



a X 



B and B in the function (1), we have Ac + A' c x + 

 A' c " '^. (Ji + /i') X-. —^ + &c. which is ultimately reduced to its 



two first terms. By pursuing a similar analysis it will readily ap- 

 pear, that if there be i roots indefinitely near to «, and representetl 

 by « + /i J «, a + /t' 5 a, « + h" 5 «, &c. ; and if we devel^pe 

 the { terms, introduced by them into (p.x, of the foum B c'-"' "^ "'-'' "^ 

 according to the powers of 3 « ; we may equate the coefficients of 



; ax V ax2»,2 k x (i — !)» (i — V) . 



, c X $ u, C X d a, , c x^ ' tf « ^ % to 



c 



