300 ME. W. SPOTTISW OODE OX THE CONTACT OF CONICS WITH SUEFACES. 
From this we may deduce the following system : 
Aft(wX-AP) + BftftY- t yP)+ . . =A^ 
A (33) 
Aft(wX-AP)+Bft(wY-yP) + . . =B=. 
Again, since AA=0, AB=0, AC=0, AT)=U, it follows that 
A A (kX- .rP) + B A(mY - yPi + C A(»Z-sP)+ D A(«T- tV) 
= — “{(3, . .)(uA~aA, ccB — pA, aC — yA, aD — 5A j( d,, B ff , 3 ; , 3,)(wX — ,rP) 
+(3, ■ -KPA-aB, 0B-/3B, pC-yB, /3D-SB)(3„ 3,, 3„ 3,)( M Y-yP) 
+(3, ■ .)(yA- a C, yB-pC, yC-yC, yB-<>C)(3„ 3„ 3,, 3,)(«Z-zP) 
+ (3, ..)(*A-«D, 5B-/3D, SC-yD, 8D-SD)(3„ 3,, 3„ 3,)(«T-<P)}. 
But since 
3A+?i; B+0C+*, D = 0 ■ 
®A+<6 B+JfC+iMD=0 ^ 
®A+jfB+CC+#lD = 0 ^ 
lA+iMB+flC+B D=0, 
the expression above written reduces itself to 
^{A(&, . .)(«, ft y, 5)(ft, ft, ft, d t )(uX— aP) 
+B(0, . .)(«, ft 7> ft, ft, ft)ftY— yP) 
+C(3, ..)(«, /3, y, &)(ft, ft, ft, ft)(t*Z-zP) 
+D(& . .)(«, ft y, S)(ft, ft, ft, ft)(wT — £P)} 
= £{9(, ..)(«, ft y, S)[(wX— AP)ftA+(?*Y— yP)ftB-j- .. 
(wX-AP)ftA+(wY-^P)ftB + . . 
: : 1 
= -^(3, ■•)(«, P, y, S)(A, B, C, D) 
= 0, 
i.e. AA(wX— aP) + B A(?/Y + yP) + CAftZ — zP) + D A(wT — tP) = 0. . . (35) 
Hence, if we multiply the first three columns of (32) by A, B, C respectively, and 
F 
add them to the fourth multiplied by D, the whole equation will be divisible by — ; and 
ZT 
on the division being made the equation in question will take the form 
ft(wX-AP) ft(wY-yP) ftftZ-zP) A u =0 . . . (36) 
ft(wX — aP) ft(wY— yP) ftftZ — zP) B v 
ft(wX— aP) d z (uY—yP) d z (uZ — zP) C w 
d t (uX— aP) d t (uY—yP) ftftZ— rP) D k 
A(uX— aP) A(wY— yP) A(uZ—zP) . vH 
which is of the degree 18w— 24. 
