U6 A NEW PROOF OF THE EXISTENCE OF 



A NEW PROOP 0¥ THE EXISTENCE OF THE ROOTS OF. 

 EQUATIONS. 



BT THE BBV. GBOEGEPAXTOBr YOUNG, M.A., TOfiONTO. 



The equation of the m'* degree, 



/(a;)=a;"'+«i«"-» + +«™=0, -^ (1) 



has a root. For, y and z being real variables, 



/0+ \/^«) = -P (cos \+ v/~ sin X) ; 



where P and A are real. When y and s receive the definite values 

 y, and r„ let P and X become P, and Xi respectively ; and let P? be 

 the least possible value of P^. Then yi + -v/^ 2^, or, as we may call 

 it, a?i, is a root of the equation, 



/ («)— P, (cos X,+ v/=^ sin X,)=0 —(2) 



Let n be the greatest number of roots equal to a;, which this equation 

 has. Then /(ic)— P, (cos Xi+ y/^ sin Xj) is divisible by (a;— ar,)** 

 without remainder : which we may express by putting 



/(a?)-P,(cos X,H- y=rsin X,) = (a;-a:,)" {F(z)] — _(8) 



Take pc<i=Xx ^Jc (cos ^ + y~ sin <^) =a;i + h. Then 



P(x8)= P(a;,) + i, A 4- Xs A* + &c. ; 



where X„ X^, &c., are clear of h. In order to separate the real from 

 the imaginary parts in the value of F {x^, put 



F {x^^—A (cos 6+ s/~ sin &)> X, = S (cos 1/^+ y=T sin «/^), 



and so on. Since equation (3) is independent of the particular value 

 of %, we may substitute ar, for x in that equation. Then 



/ (arj) = Pj (cos X, + ^-^ sin X,) + A" J P («,) + X, A + ^c } 



=P, cos X, + ;fc".J[ cos in^-^ff) + 



+ y/^ {P, sin X, + ^» ^ sin (w <^+^) + ^c}- 



