407] 
CURVES WHICH SATISFY GIVEN CONDITIONS. 
271 
(Z) 3 {(», 1)-(T, 1, 2) — (3, 1) (2m — 5)} 
+ 2 {2 (2, 2) — (I, 1, 2)-(2, 2) (2m-5)} 
4- {2(1, 1, 2)-(I, 1, 2)(2m — 6) — (T, 1, 2)(2m-6)} 
+ Supp. (I, 1, 2) = (I, 1, 2)2D; 
(Z) 2 {(2, 1,1)- (I, 1, 1, 1) 3 - (2, 1, 1) (2m - 6)} 
+ [4 (1, 1, 1, 1) - (I, 1, 1, 1) (2m - 7) - (1, 1, 1, 1) (2m - 7)} 
+ Supp. (I, 1, 1, 1) =(T, 1, 1, 1)2D. 
110. I content myself with giving the expressions of only the following supplements. 
Supp. (4X)(1) 
= m [2 (•)-(/)]+ «(•). 
Supp. (SZ) (2) 
= M2(0—(•/)]+!*(•/)■ 
Supp. (SZ) (T, 1) 
= ( 2 mn — 3 n 2 — n + not) ( : ) 
4- (2m 2 — 4 mn — 2m 4- 2n + (m — 
-1- (— m 2 + m 
Supp. (2Z) (3) 
= — \m [2 (.’.) — (: / )] 
+ \n [2 (.-.)-(:/) + 2 (2 (:/)-(•//))] 
+ \ K ('■ /)• 
Supp. ( Z)(4) 
= (Ik b (2k 4" 2t), 
)(//)• 
where a, b are the representatives of the condition Z. 
It may be added that we have in general 
Supp. (Z) (4X) = a Supp. (4X •) + b Supp. (4X /), 
where (4X) stands for any one of the symbols (4), (3, !)....(!, 1, 1, 1). 
111. The expression of Supp. (4<Z) (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 
= ^[2(3X:)-(3X-/)]; which being so, the number of the point-pairs (SZ) which, regarded 
as lines, touch the given curve (of the order m and class n) is = [2 (SZ:) — (SZ • /)]. 
The point of contact of any one of these lines with the given curve is (specially) a 
united point, and we have thus the term \n [2(3X:)-(3£- /)] of the Supplement. 
2°. The number of the conics (SZ) which touch the given curve at a given cusp 
thereof, or, say, the conics (3Z)(2/ci), is = ^(3Z • /), and the cusp is in respect of each 
of these conics a united point; we have thus the remaining term \k (SZ • /) of the 
Supplement.