116 
Proceedings of Royal Society of Edinburgh. 
£%§§).= 
1 
1 
1 
1 
a a 2 a 2, I 
b b 2 b 3 
c c 2 c 3 
d d 2 d 3 , 
1 1 a 
c 2 c 3 
|l 6 
d 2 d 3 
1 b 
a 2 
a 3 
1 c 
d 2 
CO 
1 
a 
b 2 
b 3 
1 
c 
d 2 
d 3 
1 
b 
a 2 
a 3 
1 
d 
c 2 
c 3 
& 2 &3 
c 2 c 3 
a 2 a 3 
& 2 &3 
= (b - a)(d - c)c 2 d 2 - ( c-a){d-b)b 2 d 2 + ( d - a)(c - b)b 2 c 2 
+ (c — - a)a 2 d 2 - (cZ -- 6)(c - a)a 2 c 2 + (d - c)(b - a)a 2 b 2 , 
= (5-a)(^-c){a 2 & 2 + c 2 c? 2 } 
+ (c - a)(6 - c£){a 2 c 2 + <# 2 & 2 } 
+ (c£- ^)(c - 6){a 2 c£ 2 + b 2 c 2 } . 
By Jacobi, however, the result here established is given merely 
as an example of an improved general theorem, which is enunciated 
in the form of a ‘ rule,’ as follows : — 
“ Fingatur expressio 
(a>i - ct 0 )(a s ~ « 2 ) ’ • • ( a n ~ an 
2 2 4 4 
i)2WA 
n-1 n- 
%-i a n 
“ quam quo clarius lex appareat sic scribam 
(cm - «o)( a 3 - a 2 ) • • • (°n - «n-l)2K a i)°( a 2 a 3) 2 ( a 4 a 5) 4 * • • (<^-l<hi) n ~\ 
“ sub signo % omnimodis permutatis exponentihus 
0, 2, 4, . . , n-1. 
ie In expressione ilia cyclum percurrant primo elementa tria 
Cf'n - 2j Ct n _ 1, CL n , 
“ secundo elementa quinque 
a n- 4 , a n . 3 , a n -2, a n - 1 , a n , 
“ et sic demceps itajit postremo cyclum percurrant elementa 
*%> * 
“ Omnium expressionum provenientium aggregatum sequa- 
** bitnr ipsi P.” 
