ME. W. SPOTTISWOODE ON HYPEE JACOBIAN SUEFACES AND CUEVES. 363 
It is then, in the first place, clear that in virtue of these relations we may, at the 
points in question, regard any two of the first four columns of the determinant P as, 
a une facteur pres, equal to one another ; and, consequently, any derivative of P in 
which any two of those first four columns remain unaffected will ipso facto vanish. 
Hence, for our present purpose, we shall have 
P=0, (^,^,B 2 ,^)P=0, AP=0 (3) 
Also, if we write (d, A) for (d,A, . . d a , d,) 2 , then 
B,AP=(3,A)P-f Ab,P. 
But (d x A)P vanishes for the same reason as AP ; hence operating upon the first deri- 
vative of the equation (t : m)P=P 0 +mUP 1 , viz. upon the equation 
(t : m)d x P=d J^+mzsPi+mUdJ?!, 
and putting^?, for the degrees of P, P 1? AP, APi respectively, we shall obtain 
(t : m)d x AP=AB x P 0 +m{P 1 AM+wAP 1 + 2Hd x P 1 4-4HchP 1 | 
+UA^P ) +2(p 1 -l)H3,P 1 ;(w-l)f. J ( ' 
But since P 0 contains the columns U, u, v, w , it follows that d,AP 0 must contain either 
the column U, which vanishes, or two of the columns u, v, w, any two of which have' 
been shown to be, a une facteur pres, identical. Hence 
(Ad,, AB„ &d z , AB,)P o =0 (5) 
Also, since Pj contains the three columns u, v, w, it follows that 
(^,^,^ = 0 ; ( 6 ) 
and we may therefore conclude from (4) that in the present case 
c),AP : «i=B i ,AP : / y=B.AP : w=chAP : ^=mmAP, : t ; .... (7) 
that is to say, at the secondary points of the system the Hyperjacobian surfaces have 
four-branch contact with the given surface, and consequently with one another ; and the 
Hyperjacobian curve has a quadruple point at these points. 
It does not, however, appear that the contact between the Hyperjacobian surface and 
the given surface is more than four-branched. This will be seen from the following 
process, which, although leading to only a negative result, is perhaps worth placing on 
record on account of the peculiarity of the algebraical result. 
Operating with A upon the equation (t : r/i)P=P 0 +mUPj, we obtain 
(t:m) AP+2DT : m=AP 0 +4mHP 1 +mUAP 1 +2mp 1 HP 1 : (n- 1) 
=AP 0 +2m{2+p 1 :(w-l)(HP 1 +mUAP,. 
