232 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[406 
where the terms in { } are 
— (a + 1 — a!) (a + 2) (a +3), (a + 2 — a") (a + 3) and — (a + 3 — a'"), 
that is, 
— (d 4" 1) (a + 2) (a + 3), (c + d + 2) (a + 3) and — (b -f- c + d + 3) 
respectively ; whence the whole expression is 
— f Sd (a + 1) (b + 1) (c + 1) (d + 1) . (a + 2) (a + 3) "| k, 
— %cd (c + d+ 2) (<x+ 1) (b + 1). (a+8) 
+ 2tbcd (b + c + d + 3) (a + 1) 
— 6 abed 
the expression multiplying (a 4- 2) (a + 8) is 
{a +1) (b + 1) (c + 1) (d + 1) Sd, = (a +1) (b + 1) (c + 1) (d + 1) & ; 
and we have moreover 
(a + 1) (b + 1) (c + 1) (d +1) = (1+ a + /3 + y + 3) ; 
the other lines are of course expressible in terms of (a, /3, y, 8), but as the law of 
their formation would then be hidden, I abstain from completing the reduction. 
82. The series of formulae is 
[a] 
= (a. + 1) rm 
+ (a + 1) aA 
aie, 
. rm 
[a, b] = -(a + l)(b + 1) (a +1) 
— (a + l)(i + 1) J a(a + l)] A 
1- /3/ 
+ I Si (a + 1) (i + 1)^ k, 
[— ab J 
where a = a + b, ¡3 = ai ; and coeff. of k expressed in terms of a, /3 is = a (1 + a. + ¡3) — /3. 
[a, b, c]= (a + 1) (i + 1) (c + 1) (a + 1) (a + 2) ..rm 
+ Ça + 1) (i + 1) (c + 1) ( a (a + 1) (a + 2)1 A 
< - /3 (a + 2) • 
l - 7 J 
+ f-S c(a + l)(i> + 1)(c + 1)(a + 2) ï /c, 
+ Sic (i + c + 2) (a + 1) 1 
[ — 2 abc j 
where a = a + b + c, /3 = ab + ac + be, y = aie ; and the coefficient of k expressed in terms 
of a, /3, y is — — or — or/3 — a 2 y — 3a 2 — a/3 — 2a + 2/3 + y.