Question of solving the Equatio7i of the Fifth Degree. 29 



Demonstration. — If we denote by .%\ a\ x^ x^ x^ the five roots 

 of the proposed equation of the fifth degree, and put, as is 

 permitted, 



/(xi) = hy-\-qx^,f{x^) = h^ + qx^^,f{x^) = hs + qx^,\, . 



fix^) = h^ + qx^,f{x^) = qx^, J ^''•'' 



and Q + q=Q', (8.) 



the result of the elimination of x between the two equations 

 {55.) and (2.) may be denoted thus, 



(i/-Q'-^-,-/^i) {y-Q''^\-h.2) iy-Q'^s-h) {y-Q'x,-h,) 



xiy-Q! x^) = 0; (10.) 



and if this result is to be of the form (56.), independently of 

 Q, and therefore also of Q', we must have the six following 

 relations : 



■«"l + -*"2+*3 + -^4 + '^5 = 0, (11.) 



X■^ Xc^ X2~\'Xq .^g X/^ + .fg X/^ X^ + "^4 X^ Xx -l" .i'g .J'j Xc^ 

 -VX-^X.^X^-\-X,iX^Xc,-\-X^X^X^+X^X^X^-\-X^Xci^X^= 0, (13.) 



h.^h^ + h.^^h^^Q, (16.) 



^1 (-^2 ^2, + -^2 •*'4 + -^'S -^5 + -^'a -^'4 + ■^S ■^S + -^4 ^\ ) 



+ /i2(&c.) +/i3(&c.) + /;4 (&c.) = 0, (63.) 



h^hc^[x^^rx^-\-x^ + h^ /^3 (&c.) +h^h^ (&c.) 



4-/^2^3 (&C.)+^2^i4(&C.)+A3 ^4 (&C.) = 0, (64.) 



hjh^h^-^h^hcih^ + h^h^h^ + h^h^h^= 0; (19.) 



of which the two first give 



A = 0, C = 0, (57.) 



and the three last may, by attending to the first and third, 

 and by eliminating h^, be written thus : 



h, {oc,^-x^^) +^2 {x^^-x^^) +/i3 {x._,^-xl) = 0, (20.) 



hi^Xi + h^^Xci + hs^X3+{hi + h^ + /i._^)-X4= 0, (21.) 



{h^ + h^) {h^ + h,){k, + h^) = (22.) 



Selecting, as we are at liberty to do, the first of the three 

 factors of (22.), namely, 



^2 + ^3 = 0, (23.) 



and eliminating 7^., by this, we reduce the two conditions (20.) 

 and (21.) to the two following : 



hx{xx'^-x^^) + h^(x^'>-X2^) = 0, (24.) 



^x^{x,+X4) + k^^ {x^ + x^) =0, (25.) 



which give, by elimination of 7^2 , 



h,''{x,-\-x,){{x, + x.{){x,-x,f + iXi + Xs){x^~x^y} = 0. (26.) 



