J7Q Mr- Horner on the [March, 



Secondly. Let b + c = o. Then s,^^ = (a d -' h c) s„ 

 ■whence q, + 2= q.> and therefore^ + '^ a: = /' x, and/^ x = a'. 

 Hence the solution of this equation is 



/a?= ''-^ (12) 



4. To proceed then with the general inquiry, in which both A 

 and B are effective quantities, equation (10) transposed becomes 



S.+.- As,+ . + Bs. = (13) 



and, by the theory of equations, this condition is satisfied when 



$, = ar' + &f (14) 



r, f being the roots of the equation 



y- - A y + B = (15) 



a and /3 constant quantities whose values remain to be deter- 

 mined, and z any quantity whatever. In fact, direct substitution 

 produces 



a f (r"- - A r + B) + |3 f)^ (/j'^ - A p + B) 



which is necessarily = 0, because each of the parenthetic factors 

 is so. 



Now when z = I, we have, by (1) compared with (5), a s., = 

 ab + a c, giving s, = b + c = A = r + p ; and by (5) alone 

 Si = 1. Hence the equations 



5, = ar"- + /3p"- = r + p 1 .jg. 



s, = «r + p=l J^ 



which yield by elimination, « = — |3 = — — •, and consequently 



s„ == « + |3 = 0. And these results are confirmed by making 

 s = in equation (13), which thus becomes, as it ought, 



s^ — A s, + B So = 

 The correct solution, therefore, is, 



S. = ~ .". q, — : r— (1/) 



r — p ^ r- — p- ^ ' 



This solution, detached from the investigation, but succinctly 

 demonstrated, according to Diarian usage, has been more than 

 five years before the public, and is the most extensive, perhaps, 

 that can be obtained. It remains for me to show its application 

 to some general cases. 



6. When r, p, are imaginary, we assume, as usual in quadratic 

 equations of that description, 



cos.^ = --4^= "'"' (18) 



^ VB 2 Vbc -ad 



which leads to r, p = (cos. S + a/ -- 1 sin. S) -/ B, and r% p' 

 = (cos. z^ ± V — 1 sill, s: S) V" B". 



