1899-1900.] Dr Muir on Jacobi's Expansion. 135 
write a 0 , a 4 , a 2 , a s , a 4 , a 5 in place of a , b, c, d, e, f : that having 
done so and obtained the first term 
■(a 1 -^a 0 )(a 3 -a 2 )(a 5 -a 4 y(a 0 4 a 1 4 a 2 2 a 3 2 + + ), 
he would then derive two other terms by cyclical permutation of 
the elements a 3 , a 4 , a 5 ; that next he would derive four others from 
each of these three by cyclical permutation of the elements 
a 4 , a 2 , a 3 , a 4 , a 5 ; and that the 1 5 terms got in this way must all 
be taken positive. His words are : — 
“ Fingatur expressio 
(a x - a 0 ) (a 3 -a 2 ) .... (a n - a n _^) Sa 2 2 a 3 2 a 4 4 a 5 4 .... a%z\a 7 J 1 , 
quam quo clarius lex appareat sic scribam, 
(®1 “ a o) ~ ^2) • • • • (&« “ Mu— 1) 2(®0 a l)° (® 2 * 3 ) 2 (^4 %) 4 • • • • (cin—iCt"ii) n b 
sub signo % omni modis permutatis exponentibus, 
0,2,4, n - 1 . 
In expressione ilia cyclum percurrant primo elementa tria, 
a n —2 5 oi n _ 4 , a n , 
Secundo elementa quinque 
a n - 4 , a n _ 3 , a n _ 2 , a n _ 4 , a n ; 
et sic deinceps ita ut postremo cyclum percurrant elementa 
a 4 , a 2 , a 3 a n . 
As has been stated Jacobi confined himself to a mere enunciation 
of his theorem : in fact, the two Latin sentences just given contain 
all that he has said in regard to it. 
The object of the present paper is to draw attention to a totally 
different mode of formation of the terms of the expansion, and to 
establish the accuracy of both modes. 
(3) Each term of the expansion, it will have been noted, consists 
of two parts, ( 1 ) a set of linear binomial factors, ( 2 ) a single non- 
linear factor. What we therefore require is a rule for finding the 
various sets of linear factors, a rule for deriving the single non- 
linear factor from the set of linear factors to which it is attached, 
and a rule of signs. 
How to find the various sets of linear factors we have only to 
