1905-6.] Dr Muir on the Theory of Alternants. 
367 
-K-( a 1 ? a 2 ? a 3 ? • • • > a n-l) ^ 
(a n - a 1 )(a n - a 2 )(a n - a 3 ) . . . , (a n - a^) n ’ 
m 
where by K is denoted “die ??ite Klasse der Kombinationen ohne 
Wiederholungen.” 
Rosenhain (1849). 
[Auszug mehrerer Schreiben .... liber die hyperelliptischen 
Transcendenten. bTo. IV. Crelle’s Journ ., xl. pp. 347- 
360.1 
In the course of an investigation regarding the relation between 
two Abelian integrals Rosenhain is brought up against the 
determinant 
^.11 1 
l\ ~ a i a 2 
In - 1 a n-l 
already dealt with by Cauchy in 1841, and afterwards known as 
u Cauchy’s double alternant.” The multiple integrals in question 
have to suffer transformation of the variables, and as a pre- 
liminary it is ascertained that the Jacobian 
V +'’£) 
dt 2 
and 
+ 'dt l dt 2 
^ ~dx 1 ‘dx 2 
dx n _ i 
dt n - 1 
U 
dx n _i 
= c.2, 
- 1 = D-Z 
1 
^1 ' ^2 ^2 
t 1 a 1 t 2 a 2 
In - 1 — a n - 1 
1 
In - 1 ~ a n - 1 
where C and D are specified functions of the a’s and t’s. From 
this by multiplication it follows that 
i v± 1 . -L_ _j. r= j_, 
( t j cq t 2 <x 2 t n _ i j CD 
and thence ultimately that 
v ± 1 . 1 .... 1 
^1 ^2 a 2 
’where 
^n— 1 a n-l 
_ ( - . a 2 , ■ ■ . , ■ n(< t , ^ , , , . , i n _ t ) 
• $(“2) • • • ^( a «-i) 
= • • (f-U 
