108 
ON A CERTAIN SEXTIC TORSE. 
[436 
if for shortness 
P = gh (0 + u) + hf(0 + /3) +/g (0 + y). 
Hence, substituting, 
34> + abc (9 + a) 3 (0 + /3)* (9 + 7 ) 3 (*) = - 27 (a&c) 5 (0 + S) 4 (0 + a) 2 (0 + /3) 2 (0 + y) 2 P S Q; 
which when the values a{6 A-a) 3 , b(9 + /3) 3 , c(9 + y) 3 for (x, y, z) are substituted in 
the functions and (*), will be an identical equation in 6. 
15. It is right to remark that what we require is the expression of (*), = (*]£», y, z) 2 ; 
the foregoing equation leads to the value of (*) expressed in terms of 6; and it is 
necessary to show that this leads back to the expression for (*) as a function of 
'(.x, y, z); in fact, that the function of 6 is transformable in a definite manner into a 
function of {oc, y, z). Suppose that the function of 6 could be expressed in two 
different manners as a function of (x, y, z); then we should have two different 
functions (x, y, z) 2 each equivalent to the same function of 9; and the difference of 
these functions would be identically =0; that is, we should have a function (x, y, z)' 2 
vanishing identically by the substitution 
x : y : z = a(9 + a) 3 : b(9 + /3) 3 : c(9 + y) 3 ; 
but these relations are equivalent to the single relation 
(a 2 x + b 2 y + c 2 z) 3 — 27 a 2 b 2 c 2 xyz = 0, 
which, qua cubic equation, is not equivalent to any equation whatever of the form 
(x, y, zf = 0; 
that is, the function of 9 is equivalent to a definite functi m (x, y, zf. 
16. To proceed with the reduction, I remark that we have 
( f> = (a 2 x 4- b 2 y + c 2 z) 2 
+ xyzVL 
' / 4 [a 2 / 2 ( h Y - ™<fyz + rfiz 2 ) (b 2 y + c 2 z) + b 2 c 2 (,h?y + y 2 z ) :i ] 
+ if [ffi 2 (free 2 — 7fi 2 f' 2 xz +f i z 2 ) (a 2 x + c 2 z) + c 2 a 2 (h 2 x + f' 2 z)f 
+ A 4 [c 2 h 2 {(fix 2 - 7ff‘ 2 xy +/Y) (a 2 x + b 2 y) + a 2 b 2 {(fix +f 2 y) 3 ] 
— a 2 A 4 </ 4 (b 2 fj 2 + c 2 h 2 ) a? 
— b 2 h 4 f* (c 2 h 2 + a 2 / 2 ) y 3 
, - c 2 fy (a 2 / 2 + b 2 g 2 ) z 3 ) 
where 
— Î2 = 2 (b 2 c 2 Ax a + c 2 d 2 By 2 + a 2 b 2 Cz 2 ) 
+ (My + Nz) (a*x + 2a 2 b 2 y + 2a 2 c 2 z) 
+ (Oz + Px) (2a 2 b 2 x + b*y + 2b' 2 c 2 z) 
+ (Qx + Ry ) (2a 2 c 2 j; + 2b 2 c 2 y + cz ),