.1905-6.] 
Dr Muir on the Theory of Alternants. 
361 
_ ( ft-ZftCft-M (ft -6.) 
*1- (ft - ft) — (ft -ft)’ 
(ft-ft)(ft-M (ft-ft) 
2 ~ (ft -ft) . . . . (ft -ft)’ 
the binomial factors of the numerator in the case of x r being got by 
subtracting from (3 r all the b’s in succession , and the similar factors 
of the denominator by subtracting from (3 r all the other (3 J s. 
Remembering that Binet had originally been an expert in 
working with determinants, it is not a little curious to note that 
he did not compare with these expressions for x 1 , x 2 , x 3 , . . . . 
the expressions in terms of determinants, viz. — 
«*! = 
i(^-ft)- 1 -- 
i (^2 — A2) 1 • • 
-Pi -ft)" 1 
• •<&>- ft)' 1 
fti-ftnvftr 1 -- 
(^-ftm-ft)- 1 .. 
..ft -ft)- 1 
..(ft-ft)- 1 
l (ft,- ft)- 1 .. 
• •(ft-ft)? 
(ft-ftftXft-ft)- 1 .. 
..(ft- ft)- 1 
Had he done so he would undoubtedly have reached a result which 
was not brought to light until four years later by Cauchy. 
Haedenkamp (1841). 
[Ueber Transformation vielfacher Integrale. Crelle’s Journ ., 
xxii. pp. 184-192 *] 
The transformation referred to in the title has its origin in a 
special equation of the n th - degree in y , viz. — 
X 1 X 2 _[_ .... -p Xn — 1 • 
a !~y « 2 -y ' a n~y 
and, as Haedenkamp gives the values of x 1 , x 2 , . . . , x n in terms 
of the n roots y x , y 2 , . . . , y n of this equation he may of course 
be viewed as having solved the set of linear equations — 
* See also Crelle’s Journ., xxv. pp. 178-183 (1842), and Grunert's Archiv 
d. Math. u. Phys., xxiii. pp. 235, 236 (1854). 
