1899-1900.] Dr Muir on the Theory of Alternants. 129 
used to stand for the same as C, but to concern one element less, 
viz., a nt and C" be used in similar manner, the identities at the 
bottom of the simplification are — 
C s+ i — a n C s = C s+i? 
// 
Cs+2 - (#>i + <bi-i)C s +i + a n a n -\C s = C s+2 , 
the truth of which is apparent when we remember that C 15 
C 2 , . . . are practically defined by the equation 
1 _ 1 _Ci_ C 2 + 
(x - a^)(x - a x ) .... (x-a n ) ~~ x n+1 + x n + 2 + x n+z 
It is noted also that in the determinant a C with the suffix 0 is 
to be taken as 1, and a C with a negative suffix as 0. 
CAUCHY (1841). 
[M&noire sur les fonctions alternees et sur les sommes alternees. 
Exercises cC Analyse, ii, pp. 151-159.] 
As has before been pointed out, the preceding paper of Jacobi’s 
was the last of a triad which was followed up by a similar triad 
from the pen of Cauchy. Cauchy’s first paper, which corresponds 
in subject to Jacobi’s third, comes up therefore quite appropriately 
for discussion now. 
What is really new in the first part of it concerns the finding 
of the symmetric function which is the quotient of an alternating 
function by the difference-product of the elements ; that is to say, 
in Cauchy’s notation, the finding of 
S[ ±f(x,y,z, . . .)] ^ 
(x - y)(x - z) . . . (y-z) . . .’ 
or, in Jacobi’s notation, the finding of 
y • • •) 
^ Tl(x,y,z , . . .)' 
It therefore opens with the reminder : — 
“ Une fraction rationelle qui a pour denominateur une 
fonction symetrique et pour numerateur une fonction alternee 
VOL. XXIII. 
I 
