308 MR. W. SPOTTIS W OODE ON THE CONTACT OF CONICS WITH SURFACES. 
But DH = 0 is equivalent to the system 'bjl=6 ] u, = . . i. e. to 
ro^+2AH ' = 0^1 or %H!A=0 i u —vrp 
H7(jr -j-2BH/=0 1 i; 2H'B =0^ — urq 
w r +2CH'=^,w 2H'C —& x w—usr 
■urs -j-2DH '—QJc 2H'D =6Jc —urs. 
Substituting these values of A, B, C, D, in the equation H = 0, the terms having 0, for 
a coefficient will vanish, and the equation will take the form 
(8, • 0O> ft r, s) 2 =0 (50) 
But from what has gone before it is clear that H when resolved into its factors is of the 
form x 2 — H 0i aA// 2 , where H oj u/ 2 represents the expression (45); hence putting p/^^Af/ 2 ^ 2 , 
H 
^p=p 2 — H 0 ; and consequently since H=0, p = 2<pbjp — ^H 0 , .. But<p=dr v/H 0 , the 
upper or lower sign being taken according as one or other principal tangent is the sub- 
ject of consideration. Substituting then in the equation (50), we have 
2 ,/H„^±a,H., . .)’=o. 
But since we are seeking the condition under which either one or the other principal 
tangent may have four-pointic contact, the terms of ambiguous sign must disappear ; 
and the condition required will take the form 
4H 0 (a . .)(^, . .)■+(& . opa, b,H 0 , . .) 2 =o, 
which is of the degree llw— 24, and may be compared with Clebsch’s form, viz. 
(13, • .)(dJH 0 , . .) 2 — 4H 0 O; but the comparison of the terms in <p and <b appears difficult. 
The additional condition for a five-pointic contact on the part of one of the principal 
tangents will be □ 2 H = 0; or, having reference to (20) and to the consideration that 
H = 0, the condition will be AH = 0. But the further discussion of this question 1 
postpone to another occasion. 
