260 MR. A. CAYLEY’S ANALYTICAL RESEARCHES CONNECTED WITH 
I proceed to investigate a transformation of the equation for the section with an 
indeterminate parameter X, which touches the two sections y=0, z=0. We have 
CL'^^— gv^ — 2 “h ; 
or putting for /a and v their values \/B, \/C in the second term, 
or introducing instead of X an indeterminate quantity X, such that 
we have 
«V2= (-v/IC - jF) v' i+X^. 
Also introducing throughout X instead of X, and completing the substitution of 
\/B, \/ C for jO*, V, the equation of the section touching y=0, z=0, becomes 
(aa;+%+^;z)X+3/v^ C+2\/^+\/ —ap\/ l^-X^^r=:0. 
And it may be remarked in passing, that this is a very convenient form for the 
demonstration of the theorem ; If two sections of a surface of the second order 
touch each other, and are also tangent sections (of the same class) to two fixed 
sections, then considering the planes through the axis of the fixed sections and the 
poles of the tangent sections, and also the tangent planes through this axis, the an- 
harmonic ratio of the four planes is independent of the position of the moveable 
tangent sections;” where by the axis of the fixed sections is to be understood the 
line joining their poles. 
The sections touching ;z=0, ^=0, and y=0, are of course 
ocs/ ^-\-{hx^hy-^fz)Y-{-z^%-\-^y —hp^ l+Y^^r=0 
+3/'s/ %-\-{gx-\-fy-\-cz)7j-{-\/ —cp\/ \-]r7r . w=0, 
where 
/a'+v+/''=(N/ra-(a)Y, v=x/a, 
gJ’=s/% 
The conditions of contact of the sections represented by the above written equa- 
tions would be perhaps most simply obtained directly from the lemma, but it is 
proper to deduce it from the formula for contact used in the present memoir. If for 
shortness 
where the symbol q) (+) is used in order to mark the essentially different character 
of the results corresponding to the different values of the anibiguous sign, then 
Sc<I>(-)=/(/a'+V+/'')(s(^" +>"+«"), 
MV +¥+/>')> 
+^y'(-3/)+A"/©+xyyi6+xV(K-/jT) 
-gK-/"K. 
