ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
231 
406] 
subtracting the number of the improper solutions: but this is not so when the improper 
solutions are infinite in number; the mode of obtaining the approximate formula is 
here to be sought in the considerations contained in the first part of the present 
Memoir; see in particular ante, Nos. 8, 9 and 10. 
81. The expressions for (a), (a, b), &c. may be considered as functions of rm, 1+A, 
and k, and they vanish upon writing therein rm — 0, A=0, k = 0; they are consequently 
of the form (rm, A, k) 1 + (rm, A, k)- + &c., and I represent by [a], [a, b], &c. the several 
terms (rm, A, k) 1 , which are the portions of (a), (a, b), &c. respectively, linear in rm, A, 
and k. The terms in question are obtained with great facility ; thus, to fix the ideas, 
considering the expressions for (a, b, c, d), 
1°. To obtain the term in rm, we may at once write I) = 1, « = 0, the expression 
is thus reduced to 
(a+ 1) (b + 1) (c + l)(c£ + 1) {[rm — a] 4 + [rm - a — l] 3 a}, 
and the factor in { } being = rm [rm — a — l] 3 , the coefficient of rm is 
(a + 1) (b + 1) (c + 1) (d + 1) [- a - l] 3 , 
which is 
= — (a + 1) (b + 1) (c- + 1) (d + 1). (a + 1) (a 4- 2) (a + 3). 
2°. To obtain the term in A, writing rm = 0, « = 0, and observing that 
[D] 1 = A + 1, [D] 2 = (A +1) A, [D] 8 = (A + 1) A (A — 1), [i)] 4 = (A + 1) A (A - 1) (A - 2), 
&c. give the terms A, A, — A, + 2A, — 6A, &c. respectively, the term in A is 
(tt+l)(6 + l)(c + l)(d + l) f [— a — l] 3 a . n A 
+ [— a — 2] 2 /3. l| 
+ [-a-3Jy.-l | 
+ 8.2 j 
= (a + l)(6 + l)(c+l)(d+l) (~ a (a + l)(a+ 2)(a + 3)1 A. 
- /3 (a. + 2) (a + 3) 
I + 7 
[+28 
(« + 3) 
3°. For the term in k, writing rm = 0, D = 1, and observing that [tc] 1 , [k] 2 , [k] 3 , [«J 4 
give respectively the terms k, — k, 2k, — 6k, this is 
= f-2d (a + l)(b+ 1) (c + 1) {[- a — l] 8 + [— a — 2] 2 a' }. 1 
+ 2cd (a + 1) (6 — 1) {[— a — 2] 2 + [— a — 3] 1 a" }. — 1 
-tbcd(a- 1) {[— a — 3] 1 + a'"}. 2 
. + a bed . — 6.