WHICH SATISFY GIVEN CONDITIONS. 
139 
( 2 ’ 2 \ • )=-|(3m+/) a -3(3»i+/)-9r- 8i, 
( / )=i(3m+i) 2 -3(3m+/)-8r-9^ 
: )= 6w— 4m+3;s= 5m— 3w + 3/, 
( • /)=10w— 8m+6«=10m— 8^+6/, 
(//)= 5w— 3m+3«= 6m— 4w+3/; 
( lj 3 \ • )= 2(-4m 2 + 3mw+3w 2 +28m-32w) + 3(2m+ »— 13)*, 
( / )= 2( 3m 2 + 3mw— 4m 2 — 32m + 28w) + 3( m+2w— 13);; 
• )=10w— 10m+6»= 8m— 8 w+6/, 
(/)= 8w— 8m+6*=10m— 10w+6/. 
Annex No. 7 (referred to, No. 93). 
In connexion with De Jonquieres’ formula, I have been led to consider the following 
question. 
Given a set of equations : 
a = a (viz.b = b , c —c, See.), 
ab = ab /viz. ac= ac Se c., and the like in all the subsequent equationsx 
+(ll)a.i\ +(11 )a.c, / 
abc = abc 
+ ( 12 )(a .bc+b . ac+c . ab) 
+(111) a. b, c, 
abcd= abed 
+ ( 13)(a . bcd-\- Sec.) 
+( 22)(a5 . cd-\- See.) 
+ (112)(a ,b.cd-\- &c.) 
+ 1111 a.b .c .d, 
and so on indefinitely (where the ( • ) is used to denote multiplication, and ab, abc Sec., 
and also ab, abc Sec. are so many separate and distinct symbols not expressible in terms 
of a, b, c Sec., a, b, c Sec.), then we have conversely a set of equations 
a = a (viz. b =b, c=c&c., 
ab = ab /via. ac= ac Sec., and the like in all the subsequent equations^ 
+ [ll]a . b \ +[ll]a . c, )' 
abc = abc 
= +[12 ](a.bc+b . ac + c. ab) 
+ [lll]a . b . c, 
mdccclxviii. x 
