1899-1900.] Dr Muir on the Theory of Alternants. 
105 
Assuming this law to hold in the case of n — 1 variables A ] 
A w _i, his mode of writing it being 
h A h A h J p) \\A ai A a2 A a ”~ 1 ) = Y \\A P 
1> A 2» ‘ * •'» • |i 1 A 2 n- 1/ A-ip,n-l\\ A l 
ph+a\ . a 2 
^2 
* «n-A 
• K-i)> 
he tries to show that it will hold in the case of one additional 
variable A n , the possible variation of p being ignored as before. 
To do this he changes the factor 
into 
[X X • • •> XI 
(p) 
[xx-...x-jmxx- .x-j^-x 
and the second factor exactly as it was changed in the preceding 
chapter, performs the required multiplication, and condenses the 
result. 
The rest of the chapter is occupied with the consideration of 
special cases, the lines of specialisation being exactly those 
followed in the case of the previous general theorem. Only the 
first need be noted : it is 
[X X • 
xr. 
. a+h . a+2h 
A A„ 
^a+nh\ 
_ . . a+h . a+2h 
~ *' A 1 ‘“'2 
a+(n - l)h ^a+(n+p)h\ 
The fourth chapter does not impress one favourably, although 
the author speaks of its importance in connection with later inves- 
tigations. It is almost entirely dependent on a very special case of 
the theorem of the second chapter, viz., the case where all the 
indices, except the last, of the multiplicand proceed by a common 
difference h, and where consequently all the alternants in the 
result vanish except two. In the original notation it is 
(X 
X • • - 
ftyn-p) i\ a+h a+2h 
* | A 1 A 2 
. a+(n-l)h . s 
• * * • A n- 1 A w. 
= 
ic* . 
. a+ph . a+(p+2)h 
a+nh . s+h\ 
II 1 
* • i? A p+1 
* * Xi s 
+ ||A- . 
a+(p - 1 )h a4-(p+l)h 
* * A p-1 A j> 
. a+nh . s\ 
A »-i A J’ 
Tut for convenience in what follows it may be shortly written 
~N n -p • A s = + Mp g . 
