CURVES WHICH SATISFY GIVEN CONDITIONS. 
153 
Supp.(2Z)(3) =-M 2(.*.)— (-*/)] 
+^[ 2 (.-.)-(:/) + 2 ( 2 (:/)-(.//))] 
+H-/y 
Supp. (Z)(4) =a*+6(2*+2/), 
where a, 5 are the representatives of the condition Z. 
It may be added that we have in general 
Supp. (Z)(4X) = <z Supp. (4X - )+£ Supp. (4X/), 
where (4X) stands for any one of the symbols (4), (3, 1) ... . (1, 1, 1, 1). 
111. The expression of Supp. (4Z)(1) has been explained supra, No. 108. That of 
Supp. (3Z)(2) may also be explained. 1. The point-pairs of the system of conics (3Z), 
regarding each point-pair as a line, are a set of lines enveloping a curve ; the class of 
this curve is equal to the number of the lines which pass through an arbitrary point, 
that is, as at first sight would appear, to the number of point-pairs in the system (3Z • ), 
or to 2(3Z:) — (3Z • /): it is, however, necessary to admit that the number of distinct 
lines, and therefore the class of the curve, is one-half of this, or =i[2(3Z:)-(3Z./)]; 
which being so, the number of the point-pairs (3Z) which, regarded as lines, touch the 
given curve (of the order m and class n) is =^w[2(3Z : ) — (3Z • /)]. The point of con- 
tact of any one of these lines with the given curve is (specially) a united point, and we 
have thus the term ^w[2(3Z:) — (3Z • /)] of the Supplement. 2. The number of the 
conics (3Z) which touch the given curve at a given cusp thereof, or, say, the conics 
(3Z)(2/sl), is =^(3Z . /), and the cusp is in respect of each of these conics a united 
point ; we have thus the remaining term ^(3Z • /) of the Supplement. 
Application to the Conics which satisfy five conditions of contact with a given Curve . — 
Articles Nos. 112 to 135. 
112. We have twelve equations, which I first present in what 1 call their original 
forms; viz. these are — 
First equation : 
{(5)-(5)(2m-5)-(T, 4)} 
Supp. (5) =5(5)2D. 
Second equation : 
2 { (5) (4, 1)— (2, 3)} 
+ {(4, 1)— (4, l)(2ra— 6) — (1, 1, 3)} 
+ Supp. (4, 1) 
= 4(4, 1)2D. 
