166 Mr. Lubbock on the general Solution 



The rule now is, raise 



p -\- qz + rs^+ &c. 

 to the wth power, make all powers of z equal to zero in the 

 result (in consequence of the nature of the roots of unity) ex- 

 cept such as are multiples of n, in these write n for z°, z'", 

 z^", &c. ; the result is m times the coefficient of x~"* in the 

 logarithm of the left hand side divided by .r" of the last equa- 

 tion with a contrary sign. The equations of condition thus 

 found between p, g^ r, &c. and C, D, E admit generally of 

 reductions, and in this manner uniform solutions of the qua- 

 dratic, cubic, and biquadratic equations may readily be ob- 

 tained. The same method may also be extended to the equ- 

 ation of five dimensions, so far at least as to exhibit the ana- 

 logous equations between J9, q, r, 5, t, and C, D, E. This 

 method was first given by Bezout in the Memoires de V Aca- 

 demie for 1765. 



Passing over the Solution of the quadratic eqtiation, I pro- 

 ceed to the 



Solution of the cubic equation. 



a;3 + D^ + E = (1.) 



I assume x = p + qi/ + rif (2.) j/^ = 1 (3.) 



The first equation of condition gives p = 0, which is the 

 case whenever the second term of the equation 1. is wanting. 

 I therefore take x = qj/ + r7/% 



D E 



The equations of condition in this case are —- qr = -—- 

 _ ^3 __ ^ _ E 



27 



which agrees with the well-known solution. 



Solution of the Biquadratic liquation. 

 ^ + D^ + E = (1.) 



