ME. W. SPOTTISWOODE ON THE CONTACT OF CONICS WITH SURFACES. 299 
and if, as was indicated above, H in the transformed expression is supposed to be differ- 
entiated without reference to A, B, C, D ; then 
( n — 2 )(uX — 3HP,)= — wX-f-AP 
(n- 2)(wY - 3HP 2 ) = - uY + y P 
(n—2)(uZ — 3HP a )= —uZ +zV 
(n-2)(uT — 3HP 4 )=— wT +t P, 
and the equations, of which (31 a) is one, take the form 
(wX-AP)a+(%Y-yP)h +(ttZ-zP)g+(«T-«P)l =0 
(vX-xQ)h+(vY-yQ)b +(vZ -sQ)f +(v T-*Q)m=0 
(wX— AR)g + (wY— yR)f -|-(wZ— zR)c-|-(wT — £R)n —0 
(wX-aS)I +(wY-yS)m + (wZ-zS)n+(wT-^S)d =0. 
Representing any one of these equations, say, the first, by W = 0, the equations □ 3 V=0, 
□ ‘V = 0, □ 5 V = 0 may be replaced by a system of the form (20); and writing them in 
the form = . ., where d 1 is indeterminate, we may from the five equa- 
tions so written eliminate the five quantities a, h, g, 1, 0, ; and the resulting equation 
takes the form 
B,(wX — aP) c)fuY — yF) 
3 y (wX— aP) ~dfuY—yF) 
B,(wX-aP) 'bfuY-yF) 
B^X-aP) ’dfuY-yF) 
A(mX-aP) A(uY-yF) 
where v is a numerical factor. 
'dfuZ — zb*) B.,.(wT— £P) u =0 
'd y (uZ—zF) ~d g (uT — £P) v 
~d,{uZ— zF) 'bfuT—tF) w 
B,(wZ — zF) B,(wT — tF) Jc 
A(uZ-zF) A(uT-tF) »H, 
(32) 
§ 4. Determination of the extraneous factors. 
The degree of the equation (32) in its present form is 23^—32; but it admits of re- 
duction, in the first place, as follows. Since the equation 
A(mX- aP) + B(mY —yF) + C(wZ -zF) + D(mT - *P) = 0 
is identically satisfied, we have by differentiation, and by substitution of the values of 
B,A, . . from (11), 
Ab,(MX-AP)+BB,(wY-yP)+CB,(MZ-«P) 4 -DB/mT- f\ P) 
= — (wX — aP)^ A — (uY — y~PJd — (uZ — zP)B a C — (uT — £P)B,D 
= -l{(uY-xF)(ccA-*A)+(uY-yF)(aF-pA) + (uZ-zT)(ccC-'yA) + (uT--tF)(ccT>-ZA)} 
= ~{cc (uX — aP) ■ + |3( wY — yF ) + y {uZ — zF) + X(mT — tF ) } 
A T7- 
= — K, suppose. 
2 k 2 
