74] FUNCTIONS OF THE SECOND ORDER. 431 
where, writing down the expanded values of 23, ©, Jp, p|, 
(\b + B) (Xc + C) — (\f + F) 2 = , 
(Xc + C) (Xa + A) — (\g + G)' 2 = 23 , 
(Xa + A) (Xb + B) - (\h + Hf = ©, 
(X<jr + G)(Xh + 4/) — (Xa + ^4) (X/* + 1^) = jp, 
(U + #) (X/+ JP) - (X6 + 5) (X^ + G) = ffi, 
(X/ + i^) (X^r + G) - (Xc + G) (X/i + = 
By writing successively X = —J. l5 X = — B 1 , X = — (7,, we see in the first place that 
J-u jB 1} C 2 are the roots of the same cubic equation, and we obtain next the values of 
a 2 , ft 2 , 7“, &c in terms of these quantities A 1} i? 3 , C 1 , and of the coefficients a, b, &c., 
A, B, &c. It is easy to see how the above formulas would have been modified if 
«!, b x , c 3 , instead of being equal to unity, had one or more of them been equal to 
unity with a negative sign. It is obvious that every step of the preceding process is 
equally applicable whatever be the number of variables.