168 On the general Solution of Algebraical Equations. 



of the qimntitiesyj,y^2?y3> ^^'j or the number of terms which 

 are wanting in equation 1. 



I apprehend that the equations of condition which result 

 when we attempt to transform in this manner equation (1.) 

 into the equation j/*— 1 = 0, are far simpler than those which 

 obtain when we endeavour to transform equation (1.) into the 

 equation, also solvible, 



y5 + By + -i B^^ + E = 



or indeed into any other. 

 Instead of assuming 



X — p + qy + ry^ + &c. 

 we might assume 



3/ = p' + ^' A' + r^ x^ + &c. /i/ = (3.) 

 X being raised to the ?i — 1^^ power in the last term of the 

 expression for ?/. In order to eliminate x between this equa- 

 tion and equation, (1.) we must form the product 

 Sy — j)'—q'a — r'a^ —.., &c.} 



{j^-p' ^g'b-'/b^ &c.} =fyz=0. 



And as this equation must be identical with equation (3.), di- 

 viding by ?/" and taking the logarithms as before, we may 

 find at once the equations of condition which obtain between 

 the quantities p\ q', r', &c. 71,7^,7^, &c. and the coefficients 

 of the equation (3.). Suppose for example equation (3.) is 

 j/^ — 1 = 0, we get 



P'fo + Q'fi + r'f + s\f, + t'f, = &c. &c. 

 and generally the coefficient ot y~'^ on the left hand side of 

 the equation between the logarithms will be found by raising 



{p' + q'z + r' ;^2 + s' z^ + &c.}'« 

 and writingyji instead of 3" in the result, which will bedtimes 

 the coefficient of y~"' in the logarithm of the left hand side 

 divided by ?/" of the last equation with a contrary sign. The 

 simplicity of the equations which result between p', q', r', &c. 

 will now depend greatly upon the nature of the roots of the 

 proposed equation and upon the number of terms which are 

 wanting in it. But they will, I believe, generally prove less 

 simple than those which obtain between the quantities p^ q^ r, 

 &c. and the roots of the proposed equation, and of which I have 

 given some examples. 



I have offered these remarks upon the general solution of 

 equations because it is of importance to cultivate general 

 methods. The solutions of the quadratic, cubic, and biqua- 

 dratic equations according to the method here given are as 

 simple as those, devoid of any connecting principle, which are 

 usually contained in elementary works on Algebra. 



