294 MR. W. SPOTTIS W OODE ON THE CONTACT OF CONICS WITH SURFACES. 
or 
(n— l)%vh — ivg+Jcaf^zx'-fy — 2(n~~ 1).tD(vA— wg+ka) — .BCD 
B v x v! to' 
C vl w x n' 
D ml vl k v 
Similarly, it will be found that 
(n — 1 )\wf- uli -\-pb)(ug — vf-\- kc ) 
—yz<^— 2{n— l){yU(ug— vf-\-kc)-\-zn(wf— uh-\-1cb)}— . A C D 
A u x v ' l' 
- 
B vl w x ml 
D l' ri k x ; 
( 15 ) 
(16) 
so that, m being the degree of V, we shall have, on collecting all the terms of the forms 
(15) and (16), the following expression: — 
(n — l) 2 { (vh — wff -f kafbl -f- . . + 2( wf— uh — kb) ( ng — vf-\- kcfifi . ■ -f- . . } V 
=z — 2(n— 1)(to— 1){ D(vh— wg-\-kaJb x Y . .}— . BCD d 2 V— .. 
B v x v! ml 
C vl w x n' 
D ml tI k x . 
And if to each side of this equation we add 
{n—lf {(vh—wg -\-kayb' x -\- . .} {(vh-wg-\-ka)~d x V + . .} = — (n— 1) 2 {B X Y n{vh— wg-\-ka )-\- . .}, 
in which the operations d*, . . affect the quantities u, . . only and not d^V, . . ; then we 
shall have the full expression for n 2 V, viz. 
-(n-1 2 ^l + 2 ^£^){^VD(^-^+^)4-..}- • B C DdlV-.. 
B v x v! ml 
C v! w x n' 
D ml Til k x . 
Now referring to (5), and calling the ratios therein contained 0, and substituting 6u 
for d,,.V, 6v for d y V, . . , we have 
un(vh— wg + ka) -\-vn(wf—u!h- f- kb) +wU(ug— vf-\- kc) □ ( — ua —vb — wc) 
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