THAT EVERY ALGEBRAIC EQUATION HAS A ROOT. 329 



And, in order that this Remainder of the original equation-function may = 0, we must have 



sin Q 



24. In the general case of discovery of corresponding values of p and Q which make 

 A <= 0, B = 0, (discussed in the former Memoir), will have a value which is not = 

 or = a multiple of tt. The two equations just found will then be satisfied by the same 

 values of p and 6 which make 



A = 0, B-0. 



And thus, in the general case, the conditions of divisibility, derived entirely from division of 

 the equation-function by the quadratic trinomial, without any reference to imaginary quantities, 

 are the same as the conditions for satisfying the equation by the substitution of a quantity 

 partly imaginary. 



26, The case of = requires a special examination. Suppose that it is found by any 

 tentative process that the combination of the values, /» ■= i?, = 0, satisfies the equation A = 0. 

 Since cosd, cos 20, &c. are each = 1, this is the same as saying that 



R^ +p.E' + q.R^+kc. + z=0, 



or that E is a root of the equation a?' +p.ai' + q .od^ + &c. + ss = 0. The equation B = will 



be satisfied identically, because (when 9 = 0), sin 6, sin 20, &c., are each = 0. But it does not 



B 



follow that , which then takes the form - , is = ; and therefore it does not follow that 



sin 9 



the original equation-function is divisible by w^ — 2R . cos O.a; + R^ without remainder. In 



order to find the condition of this divisibility, we must find the value of -; — - when 0=0. 



•' sin0 



Since —. — -- ultimately = n, it is evident that — — - ultimately = 8 .i?' -|- 7p . R' + 6q.R^ + &c. 

 sine0 ■^ sm9 r ^ 



Consequently, that the equation-function may be divisible by x^ — ^Rx + R^, or (a? - if)', it is 



necessary that R satisfy these two equations 



R? +P.R'' ^q. R^ ^kc. + z = 0, 



8R^ ^-Ip-R' + 6q.R' + &c. = 0. 



These, it is well known in other ways, ar6 the conditions for an equation having two equal 

 roots R. 



26. Reverting now to the general investigation ; if p, q, r, &c. as far as w, are all = 0, 

 while z has a positive value ; then the equation is 



afi + z = 0; 



and the conditions A = 0, B = 0, become 



p^ .cos 89 + 2 = 0, 

 p^ . sin 80 = ; 

 from which cos 80 = — 1, p^ = z, 



4:2— i 



