AND ON NEWTON'S METHOD OF CO-ORDINATED EXPONENTS. 617 



In the next process, the numerators are all of the form m — (m +), zero or negative: the 

 next solution, and all which follow, are therefore inadmissible. We have then 9 forms of U, 

 and no more. The values of u t , real or imaginary, are obtained from one or more equations, 

 whose united degrees amount to 9. And the fact of fi^u having value, as before, determines 

 U , &c. univocally, unless indeed some of the values of w / should be equal, in which case the 

 phenomena just described are repeated. This appearance of equal roots may continue to any 

 extent : thus, having only two values of u equal, the two values of u t may be equal, and so 

 on, say up to u tl/ , the two values of which are different ; giving for y a double form such as 



in which the illustration hardly requires different symbols for the two values of u ir u p , &c. 



Before applying this to an example, for the determination of singular points, I take the 

 explicit problem of infinitely small alteration in the coefficients of an equation having equal 

 roots. Let A n y n + ... + A = be an equation having 9 equal roots, with a for their common 

 value ; and let us consider 



(J. + B n dx 1 ") y" + (4,_, + £„_, dm**) y"~' + ... + (A + B Q dah) = 0. 



Substitution of y + a for y will convert this into C n y" + ... + C = where Cg.,, ... C are 



infinitesimals, and possibly some of the others, but not Cg. Lagrange's theorem of 1776 gives 



an account of the roots which replace the equal roots, and an approximation to their values, 



by expression of y in ascending powers of da;. For example, suppose the only change made 



is that of A Q into A + dx. The transformed equation is C n x n + ... +CqW 9 + dx = 0; where 



C„ = A n , Cg does not vanish, though intermediates may vanish. The 9 roots under inquiry 



1 

 are of the form y = (dx)o (u + U), where Cgu e + 1=0: and not more than two can be real. 



As another case, let the two last terms be changed into A x + B l dai h ^ and A + B da) b °, so that 



the transformed equation is 



C n y n + C,_,y- 1 + ... + C e y e + B x dx b iy + {aB.dw^ + B dx h °) = 0. 

 The results are as follows : — 



(1) 6 < 6, , 9 (6 - h) > K O ne real root y = efaA>-*' {u + U), where B + B : u = 0. 

 And 9-1 roots of the form y = daA : ( 9_1 ) (u + U), where B x + CgM 6-1 = 0. Not more than 

 three of the altered roots can be real. 



(2) 6 <6,, (& - 6,) = 6 . There are 9 roots of the form y = dx b »-^ (u + U), 

 where B + B v u + Cgu 9 = 0. Not more than two can be real. 



(3) b < 6, , 9 (b - ftj) < fe . There are 9 roots of the form dx b » : e (u + U), where 

 jB + Cgw" = 0. Not more than two can be real. 



(4) 6 = or > fej. There are 9 roots of the form y = dx b '- e (u + U), where 

 aBt + Bo+Cg u e = 0, or a B x + C e u e = 0. Not more than two can be real. 



When singular points are to be determined, we first examine (p (x, y) =0, <j), = 0, <p y = 0, 

 and having ascertained the simultaneous solutions, if any, of the three, it will generally be 



79—2 



