358 ME. W. SPOTTISWOODE ON H YPEE J ACOBI AN SUEEACES AND CUEVES. 
But in this case we may carry the question of the contact of the Hyperjacobians a 
step further. In fact, bearing in mind that if p, q be any two rational, integral, and 
homogeneous functions of x, y , z, t, the nature of the operation A is such that, if we put 
.... (8) 
then 
Apq=pAq-\-qAp+2(&, 35, . .)(d x p, ^ y p, d z p, d t p)(d x q, d y q, d z q, d t q). . . (9) 
This being so, if we put 
=A', " 
+&>. +^=b', v nm 
«a,+dra,+ca.+^=a, > 1 j 
£Vh»,+;§R+23^=D', _ 
it follows that 
A'U : tf=BTJ : y= C'U : z=D'U : t=H(n-l) (11) 
If, then, we operate with A upon the equation ( t : m)T=T 0 -f-mUT 1 , and put r l for the 
degree of Tj in x, y , z, t, we shall obtain the following result : — 
(t : m)AT+ 2D'T : m= AT 0 + 4mHT + 2mr 1 HT 1 : (n- 1). 
Substituting for D'T from the equations d x T : u=. .=mmT, : t, we find 
(tf:.m)AT=AT 0 +2m{-l:(n-l)+2+r 1 :(»-l)}HT 1 . . . . (12) 
But if r represent the degree of Tj it is easily seen that , or 
2ri + 2w— 4 = 2(r— 1) ; so that the coefficient of HTj will be =2-)-2(r— 1) : (n— 1). 
Again, omitting terms which vanish with U, <£>, \f/, and, for brevity, writing down only 
the first line of each determinant, we find 
AT 0 — 4H, u, v, w 
4-2 u, A!(u , v, w) 
4-2v, B'(u, v, w ) 
4-2 w, C'(u , v, w ) 
4-2&, D'(w, v, w), 
where the operations A', . . are supposed to affect all the columns which follow them ; 
thus : 
u, A'(ti, v, w)=u, A 'u, v, w 
4 -u, u , A'v, w 
4 -u, u, v, A!w. 
This being so, 
AT 0 = 4H, u, v, w 4 
4- 2D \lc, u, v, w) 
— 8H, u, v, w > 
= — 4Hm, v, w 
+ 2D'T. 
( 13 ) 
