MR. W. SPOTTISWOODE ON THE CONTACT OF CONICS WITH SURFACES. 291 
a, 
a', u , 
= □ V = 0, . . . 
. . . (6) 
(3, 
ft', v, 
V, 
y\ w, 
i. 
y, &, 
where a, ft, y, b ; a', ft' y', b' are arbitrary quantities to which any values may be given, 
provided only that all the determinants of the matrix 
a, (3, y, b 
«'» (3', y', 
do not simultaneously vanish, since in that case the equation (6) would become nuga- 
tory. Comparing (6) with the equation 3,11 = 0, and observing that the differentials 
dx, dy, dz , dt may be replaced by the determinants 
(?) 
a, ft , 
r> 
b i: 
ft', 
y 
u, v , 
w, 
k || 
(the usual rule of signs being adopted, viz. the columns being taken in the cyclic order, 
the determinants are to have the signs +, — , + , — , respectively), it is clear that the 
equation (Bj+B 2 )U=0, when combined with the corresponding equation in V, is equi- 
valent to □ 3 V=0. Similarly, the equation (d 3 + 3d,cb + c) :j )U = 0, combined with the 
corresponding equation in V, is equivalent to □ 3 V = 0 ; and so on. Hence the series of 
conditions comprised in (3) may be expressed as follows: 
v=o, ny=o, □ 2 v=o, n 3 v=o, n 4 v=o, n 5 Y=o, ... (8) 
which correspond to equations (3) of my paper on Sextactic Points above quoted. 
§ 2. Preliminary Transformation. 
The next step is to effect a transformation of the first three equations of the system (8), 
corresponding to that given in § 1 of the same paper. As was stated in the introduc- 
tion, the transformation does not throw out as factors any lineo-linear functions of the 
arbitrary quantities and the variables ; but it reduces the expressions transformed to 
functions of such lineo-linear functions, viz. zj, and nr'. 
Taking the columns of the matrix (7) two and two in the usual cyclic order, viz. 
(3, y ; y, a; a, (3 ; a,b; (3,h; y,b, and calling the determinants so formed a , b, c,f g, h ; 
i. e. writing 
(3, y, i 
(3', y', V 
2 q 2 
= a, b, c,f, g, h; . 
(9) 
