1905—6.] Dr Muir on the Theory of Alternants. 
363 
r—m +l , ; . . , rt 
Vl > «2 > • • • » a n) = T( a i » «2 > • • ■ >0 • SKm » a «+2 I • • • » «nj • n(Or “ a s) 
s=l ,2 m 
appears in the form 
a lt a 2i . . a n ) = Tl{a Xi a 2 , . . . , a m ) • n(a m+1 , a m+2 , . . , , 
where the mode of denoting the rectangular array of differences 
cannot be commended. 
Sturm (1845), Terquem (1846). 
[Cours d’analyse de PEcole Poly technique, 4to, lithogr., Paris.*] 
[Sur la resolution d’une certaine classe d’equations a plusieurs 
inconnues du premier degre. Nouv. Annettes de. Math., v. 
pp. 67-68, 162-165.] 
Employing the method of “undetermined multipliers ” Sturm 
here supplies the want left by Prony, namely the solution of 
a r 1 x 1 + a^ 2 + • • • + ar n x n == l )r (r = 0 , 1 , 2 , . . . , n - 1) 
The said method may be generally described as making the solu- 
tion of a set of n equations dependent on the solution of a set of 
n — 1 equations, the latter set being related to the former in 
having its determinant conjugate to a primary minor of the 
determinant of the other set. Thus the given set being 
aqaq + d 2 x 2 + a 3 x 3 + a 4 x 4 = a 5 " 
h x i + \ X 2 + \ X 2 + b 4 X 4 = h 5 ^ 
e i^i e^x 2 -I- c 3 x 3 -I - C 4 x 4 = c 3 
d 4 x 4 + d 2 x 2 + d 3 x 3 + d 4 x 4 = d 5j 
where the suffixes are seen to run twice from 1 to n. Another identity, just 
as worthy of note, is 
r=vi+ 1 , .... n 
f *(«! , «2 a n) = tH a i • TUflr - a s ) . (’(% , a m+ i , . . , a n ). 
6‘— 1 , 2 , . . . , m - 1 
The one is exemplified by the partition 
a 2 -a x a 3 -a 4 a 4 - a 1 a 5 - a x a 6 - a x 
$3 CL 2 CL 4 CL 2 CL^ CC^ CC^ ”■ CL 2 
: a 4 - a s a 5 — a 3 a 6 — a 3 
a 5 — a 4 ~ a 4 
a 6 ~ a 5 > 
the other when instead of this the right- to-left dotted line is made to separate 
the third row of differences from the second. The former is that to which 
we have drawn attention when dealing with Jacobi’s memoir of 1841. 
* Notdhe posthumous book with this title edited by Prouliet and published 
in 1857. 
