258 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[406 
Annex No. 7 (referred to, No. 93). 
In connexion Avith De Jonquieres formula, I have been led to consider the following 
question. 
Given a set of equations : 
a 
= a 
(viz. b = b, c = c, 
<fec.), 
ab 
= ab 
(viz. ac = ac. 
, &c., 
4-( 11) a. b 
\ 4- (11) a. 
c, 
abc 
= abc 
+ ( 12)(a. be + b.ac+c.ab) 
+ ( 111) a .b . c, 
abed = abed 
4-( IS)(a. bccl +&c.) 
+ ( 22) (ab . cd + &c.) 
4-( 112) (a. b. cd 4- <fec.) 
4- (1111) a .b .c.d, 
and so on indefinitely (where the (•) is used to denote multiplication, and ab, abc, <fec., 
and also ab, abc, «fee. are so many separate and distinct symbols not expressible in 
terms of a, b, c «fee., a, b, c «fee.), then we have conversely a set of equations 
a 
ab 
abc 
a (viz. b = b, c = c <fec.), 
ab /viz. ac = ac &c., and the like in all the subsequent equations’ 
+ [ 11] a.b \ + [11] a . c, 
abc 
4 [ 12] (a. be +b.ac4-c.ab) 
4- [ 111] a. b . c, 
abed = abed 
4-[ 13] (a. bed 4-&c.) 
4- [ 22] (ab . cd 4- &c.) 
4-[ 112] (a. b . cd 4-<fec.) 
4-[1111] a.b.c.d,