1905 - 6 .] Dr Muir on the Theory of Alternants. 
371 
repeating the first set of operations he removes the factors cq - « 3 , 
a 2 — a 4 , . . . , and so on. 
After this an application is made to the solution of a set of 
linear equations which differs from Prony’s set by having 
z°, z l , z 2 , ... in place of z 0 , z lt z 2 , . . . , and where therefore, 
as Cauchy in 1812 had pointed out, the numerators of the 
unknowns, as well as the common denominator, are resolvable 
into binominal factors. The determinants in s 0 , , s 2 , . . . , 
got by multiplication, are also given. The remaining pages (77- 
84) contain illustrations. 
Joachimsthal (1854, May). 
[Bemerkungen liber den SturnTschen Satz. Crelle’s Journ ., 
xlviii. pp. 386-416.] 
In the course of his investigations Joachimsthal evaluates (§ 5) 
the determinant 
where s q =x\ +x\-r x\. Using the fact that by reason of the 
trinomial elements the determinant is partitionable into twenty - 
seven determinants with monomial elements, he shows next that 
all of the twenty-seven except six vanish ; that the six contain 
the common factor 
(* - x l)( x - x l)( x - X S ) ■ (*3 - X l)( X 3 - x tt)( x 2 - x l) i 
that the aggregate of the cofactors is 
x 2 x 2 - x\x x + x\Xi - x\x z + x lx 3 - x\x 2 
or 
0*3 ~ • r l)( ; *'3 “ X 2)( X 2 ~ X l) > 
and that therefore finally the given determinant is equal to 
<j (x 3 - x^(x 3 - x^ipc^ - x Y ) | . (x - xf)(x - x 2 )(x - x Q ). 
This is followed by the assertion that if s q were made to stand 
for + . . . +sj n the determinant could be partitioned into n 3 
