ME. W. SPOTTISWOODE ON HYPEEJACOBIAN SEEPAGES AND CUEYES. 365 
But since col. w=x col. u, it follows that 
A' col. w = col. u . A'A+X A' col. u ; 
so that the expression A! {u, A (u , . . .)} will vanish. The same will obviously be the case 
with the results of B'j and C' on the same expression ; and we may, in fact, conclude as 
follows : — 
A'{u , A (u , . . .)} =0, 
B '{u, A (u , . . .)}=0, 
C'{u, A {u , . . .)} =0, 
T)'{u , A (u , . . .)}=D'AP. 
Moreover, 
4H, A (u , . . .) = {4H, A(«, v , w), nl, nJ, nK} 
= (4— n)H{ A(a, b, c), L, M, N} 
=(4-n)HA{«, b, c, L, M, N} 
=(4— n)HAPj. 
Hence, finally, 
A 2 P 0 ={ — 1 — 4+n+2mm: (n— 1)}HAP 1 ^ 
= {-l+(2m-2-3w + 3+2w-2m):(w-l)}HAP 1 l .... (11) 
= — 2HAP,. J 
Collecting the various terms, we find 
(^:m)A 2 P=2{-l+m[4 + (p 1 +y i -2):(w-l)']}HAP 1 . 
But since 
jp' = degree of AP , .-.jp' = 2 n-\-]p —6, 
p 1 =degree of P! , pi — — n-\-]p + 1, 
p' x =degree of AP n jp\— 2 w+p,, — 6, 
consequently 
i>i+yi=2p— 4, 
Pi+p \— 2 + 4(w— l)=2 j p' + 2, 
(4j/+4 )(w— m) : m(n— 1) — 2, 
= {(4p'+4)(w— m)— 2w?(w— 1)} : m(n— 1) 
= {[4p'+4— (2n— 2)](w— m)+2(w— l)(ra— 2m)} : m(»— 1) 
= {[l-f 2Q?'— 1) : (n— 1)]+[1 +2(^ — 1): (w— l)]}m=2(w— 2m) : m. 
Hence, in the case considered before, viz. where n=2m , 
(t : m)A 2 P=(n'+n)HAPj, (12) 
where n'=l+2(^'— 1) : (n— 1); «. e. n' has the same relation to AP that n has to P. 
But since n can in no case vanish, it follows that the Hyperjacobian surfaces of a four- 
fold pencil cannot in general have more than four-branch contact with the given surface 
at the secondary points of the system. 
