CUEVES WHICH SATISFY GIVEN CONDITIONS. 
151 
= (4Z • ); and the cusp is in respect of each of these conics a united point; we have 
thus the term «(4Z • ), and the Supplement is thus =m{2(4Z • )— (4Z/)} -f «(4Z • ). 
We have moreover (4Z)(1)=(4Z • ), 2D=w— 2m+2+« ; and substituting these values, 
we find 
(4Z)(1)= (4m-2)(4Z . ) 
- m{2(4Z . ) — (4Z/)} — *(4Z . ) 
+(w— 2m+2+;s)(4Z • ) 
= n{ 4Z • )+m(4Z/), 
which is right. 
108. It is clear that if, instead of finding as above the expression of the Supplement, 
the value of (4Z)(l),=w(4-Z ■ )+m(4Z/), had been taken as known, then the equation 
would have led to 
Supp.(4Z)(l)=m{2(4Z . )-(4Z/)}+*(4Z • ); 
and this, as in fact already remarked, is the course of treatment employed in the re- 
maining cases. It is to be observed also that the equation may for shortness be written 
in the form 
(4Z) |(1) — 2(2m— l)(l)j- 
+ Supp. (1)=(1)2D ; 
viz. the (4Z) is to be understood as accompanying and forming part of each symbol ; 
and the like in other cases. 
109. We have the series of equations 
(4Z) { (1) — (I)(2m — 1) — (f)(2m— 1)} 
+ Supp. (1) =(1)2D; 
(3Z) { (2) — (2) (2m — 2) — ( 1 , 1)} 
+ Supp. (2) =2(2)2D ; 
(3Z) 2{(2)-(T,l)-(2)(2m-2)}_ 
+ {2(1, 1)-(1, l)(2m— 3)— (T, l)(2m— 3)} 
+ Supp. (T, 1) =(I, 1)2D; 
(2Z) { (3) — (3)(2m— 3) — (1, 2)} 
+ Supp. (3) =3(3)2D; 
(2Z) 2{(3)-(2, 1) — (2, 1)} 
+ {(2, 1) — (2, l)(2m-4)-(T, 1, 1)2} 
+ Supp. (2,1) =2(2, 1)2D; 
(2Z) 3 {(3) — (I, 2) — (3)(2m — 3) } 
+ {(1, 2)— (1, 2)(2m-4)-(l, 2)(2m— 4)} 
+ Supp. (1, 2) =(I, 2)2D ; 
