of Algebraical Equations, 167 



I assume x = qy + ry'^+ s y^ (2.) ^ — 1 =0 (3.) 



The equations of condition in this case are 



r^-^2qs = — 4 j^r — 4 rs^ = D. 



^ + 6 g^ s- + l^qr"^ +r^ + s^ = _ E. 



These are the equations given by Bezout, Coiirs de Mathi' 

 matiques, vol. iii. p. 220. 



q^ + s^ = , and after reductions 



^ 4 r 



(?' + s^f - \6q\s^ = E :^ - 4 r'l = E. 



E D^ 



/•'^ H ?-^ — — = 0, from which r^ is found by the 



4 64 



solution of a cubic, q and s can then be easily obtained. 



' Proceeding to treat in this manner the equation of Jive di- 



mensiofiSj 



^^+ Ca;^ + D^ + E = (1.) 



I assume 



X — qy -\- ry^ + sif + ty'^ (2.) f'—l = (3.) 



The equations of condition in this case are 



qt ■{■ rs = 



Q 



q^r + qr"^ + rf^ + s^t = -'— 



4 D 



4!fr + 6q'^t'^ + Q4^qrst + 4fqs^ + 4^rH + 6r^s^ = — 



o 



q^-\-20(f St + 30 q^r^t+ 30 q^rs^ + 20 qr^s+r^ 



+ 20 q r t^ + 30 q s'^ t"^ + 30 rs'^t + 20 rs^t+s^ + t^ = — E. 



The two last equations admit of reduction, and finally the 

 solution of the equation of five dimensions may be said to 

 depend upon the determination of the quantities q, r, 5, t from 

 the following equations : 



C 



qt + r s = q^ s + qr'^ + rt^ + s^t = ~ 



q^r + 3 qr st + qs^ + 7'^t = — 



o t C 

 g^— 20 qr^s — 2Qrs^ t + r^ + s^ + t^ = _ E — ^— — . 



5 



The same method would serve to transform equation (1.) 

 into any other equation /?/ = of the same number of di- 

 mensions, but the simplicity of the equations of condition 

 which are finally obtained, depends upon the nature of the 

 quantities Sj, Sgj ^3> ^^" ^"^ scarcely at all upon the nature 



