276 
MR. A. CAYLEY’S ANALYTICAL RESEARCHES CONNECTED WITH 
Or, forming the value of and substituting 
f'(i-jy+r(i-!y+(f-g)’(i-j)' ■ 
=M-(f-S)’+h(f+g)-^T 
which may be verified without difficulty, and thus the construction for the resultors 
is shown to be true. 
§8. 
Several of the formulae of the preceding sections of this memoir apply to any 
number of variables. Consider the surface (i. e. hypersurface) 
-^2gzx-\- ‘Ihxy 
and the section {i. e. hypersection) 
{aK-\-hgj-\-gv ...')x-\-{hX-\-hi/j-^fv {g'k-\-fi/j-\-cv . ..)z... +\/ — p'V t — 0, 
where 
'7^—a\^-\-bi/j^-\-cv^-{-2fyjv-\-2gv\-\-2hXgj... — K, 
the condition of contact with any other section represented by a similar equation is 
oKh! + bg^gJ -{- cvv' -{■f{gjv'-^gJv)-\-g {vT! -|- v'A) -|- A (\gJ + l!gj) . . . + K = VV 
where K is the determinant formed with the coefficients a, b, c,f, g, h, ... And con- 
sequently, by establishing all or any of the equations X=\/% = \/33, v=^ C, ... 
we have the condition in order that the section in question may touch all or the 
corresponding sections of the sections x=0, y=Q, ;z = 0, ... 
n 1 
Let n be the number of the variables x, y, z.., then K = 
^ 1 . 
1 B jr 
# JT C 
also }^~^{{aK-\-hgj-\-gv..)x-\-{Iik-\-bgj-\-fy..)y-{-{g'k-\-fgj-\-cv..)z...) 
= — X y z .. 
X ^ ^ ^ 
m ^ f 
V 4f C 
whence also 
K”"^(v*+K)=- 
X [/j V .. 
or 
1 X gj V .. 
X a 1 6 
X a ^ 
1 35 JT 
^ m ^ f 
V # C 
V (S JT C 
