Dr Muir on General Determinants. 
695 
1907-8.] 
sostituzioni binomie, quadrinomie, sestinomie, ecc. che occorrono per 
passare da una disposizione alV altra.” 
Baltzer, R. (1857). 
[Theorie und Anwendung der Determinanten, mit Beziehung auf 
die Originalquellen, dargestellt von Dr Richard Baltzer . . . ; vi -f 
129 pp. ; Leipzig. French translation by J. Houel, xii + 235 pp. ; 
Paris, 1861.] 
The good qualities spoken of above as belonging to Brioschi’s text-book 
are still more conspicuous in the German text-book of three years later, but 
the historical footnotes in Baltzer’s give it special value. The theory is dealt 
with in eight little chapters or sections, and the so-called applications in ten ; 
several of the latter, however, might quite well have been classed with the 
former, as they are merely concerned with determinants of special form. 
The first section corresponds closely in subject with Bella vitis’ appendix : 
and in connection therewith may be noted Baltzer’s remark (§ 2, 3) that 
any term got from the diagonal term by substituting k x , k 2 , ... , k n for 
the second suffixes 1, 2, ... , n may also be got by substituting 1, 2, ... , n 
for k x , k 2 , ... , k n in the set' of first suffixes. 
Brioschi’s mode of proving Sylvester’s theorem of 1839 is improved upon 
(§ 3, li) by taking Q one order lower than P, and using the multipliers 
0Q/0&11, 3Q/35 21 , . . ., SQ/^n-i.i o n the identities 
0P 
0P 
0P 
*> 
= 0 
Vi 
+ 
a i2V~ 
OCtn ' i , 
+ . . 
• • + a \ n ~ 
oa nn 
0P 
0P 
0P 
p 0 
Av 
da nl 
+ 
+ . . 
. . 4- 5 
OCtnn 
► 
0P 
0P 
+ . . 
0P 
= 0 
l n - l , H 
0«»i 
+ a n - 1, 24 
Mi 2 
. . + a n _ i > n - — 
0 d nn 
the result of addition then being 
+ a , 
n z\i 
2 dh 
+ . 
+ ( a in~.j + a 
db n 
, 0Q 
' 21 06 21 
+ . . 
• • +«n- 
0Q ' 
/0a»i 
, 3Q 
+ 
4- a 
0Q ' 
v 0P 
22 0A 21 
• • “ 
_ 1 ’X- l.V 
>da n2 
f SQ 
+ 
4- a 
0Q ' 
)~ 
db 2l 
. . T M'n- 
which, if we bear in mind what single determinants the expressions in 
brackets stand for, is seen to be Sylvester’s theorem in its alternative form 
