for the Imaginary Roots of an Equation, tyc. 99 



bring us nearer to the point at which the roots corresponding 

 to a and b are separable by means of unital reductions. 

 When this point is arrived at, we are at once enabled to as- 

 sign the true values of the roots by means of a continued 

 fraction, similarly to the method employed by Lagrange, as 

 the following example will show. 

 The given equation is 



^+7^4—144.^3 + 611 x*- 928X + 362 = 0, 

 from which we get successively, 



ar 1 5 +12.r 1 4 -106.r 1 3 + 231.r 1 2 — 105^ — 91 = {x x s* #-1) 

 x u s + 17 x u 4 — 48 ^ n 3 -5 x n * + 92 x n — 58 = (x n = x — 2) 

 x n f + 22x m 4 + 30# m 3 -37.r m 2 + 1 1 * m -l =6(a? in = x-3) 



at the next transformation we shall evidently lose three varia- 

 tions; taking therefore the reciprocal equation and reducing, 

 we have 



yi b -6yf + 3y l s + 25y i *-10y l -26 = o(y 1 = y-l,y=~). 



Since this equation retains the three variations, there is every 

 probability, by Budan's criterion, that the indicated roots are 

 all real. Proceeding with the reductions, and retaining the 

 same notation, we obtain 



yn—y\ \ 4 ~ l1 Vn + 8 #n 2 + 30 #n — * 3 — °— 3 variations. 

 yin+*I/in-5y lu 3 -21y n *+Uy ul + 14 ! = 0...2var. 



so that there is a root of the equation in y between 2 and 3. 

 In continuation, we have 



j/iv 5 + 9#iv 4 + 21yiv 3 — 23/iv 2 — 22#iv + 7 = 0... 2 variations, 

 and in the equation in y v there will be no variations. Again, 



therefore, we take the reciprocal equation in z = — and 



yiy 

 continue the reductions : 



7V + 13*i 4 — 20zj 8 - 4 "7z 1 2 -8s 1 + 14. = 0...2 variations; 

 from which it still appears that these roots are not imaginary. 

 Proceeding as before, we get 



7 z„ 5 + 48 * n 4 + 102 z n 3 + 41 2 n 2 — 75 * n — 4-1 = ...1 var. 

 and the equation in z UJ will contain only permanences; one 

 root therefore of the equation in z is between 1 and 2, and 

 the other between 2 and 3. 



In order to determine the actual approximate values of these 

 roots in the original equation, we have, by making y = 2*5, 

 z = 1*5, and z = 2'5, the three continued fractions, 



H2 



