1899-1900.] Dr Muir on the Theory of Alternants. 
103 
(a;*, a £ 
h 
9 . 5 
...0“ 
. a f h . a+'lh . a+nh \ 
A, A, .... A„ ) 
A“ A“ +h A 
12 p 
1 — 2 .... — n 
a+(p-l)h a+(p+l)h 
P + 1 
. . . . A 
a+nh 
)• 
The third is where the series of indices consists of two pro- 
gressions proceeding by the common difference h, and where, (of 
course, there are fewer vanishing terms in the product. 
In the next chapter the subject matter is quite similar : in fact, 
the only difference is in the constitution of the multiplier, which 
is more extensive than before by reason of the fact that in 
forming the ^i-ary combinations there is now no restriction as to 
non-repetition of an element. Thus, instead of the example 
|a r 6V| . (a h b h + a h c h + b h c h ) 
we should now have 
\a r b s c f \ . (a h b h + ct h c h + b h c h + a 2h + b 2h + c 2h ) . 
The method followed is exactly the same as before. Three simple 
cases are carefully worked out, viz., 
\a r b s \ . (a 2h + b 2h + a h b h ) , 
\a r b s c f \ . (a? h + b 2h + c 2h + a h b h + a h c h + b h c h ) , 
\a r h s c t \ . ( a? h + b 3h + c 3h + a 2h b h + a 2h c h + b 2h a h + b 2h c h + c 2h a h + c 2h b h + a h b h c h ) , 
the results in Schweins’ notation — where the change to rectangular 
brackets should be noted — being 
IX, a 2 ] (2) . ||a“X) = | a;" + “'a“ 2 ) + || JpA^" 2 ) 
Xa:] ( 2 ) .}a|a:x)= ix 
2h+ai . a 2 . a 3 \ 
■2 Ag J 
A>f + X 3 
) 
+iA»f + " s ) +ix +a x + x 8 ) 
+ |;Ai + * 1 aX + “ 8 ) + |XX + “ s A 3 + “ s ) 
A h A h 1(3) I: . aj a 2 . a 3 \ Ei . 3/M-ai ci 2 . a s \ , I 
a 2 .a 3 J .|a a ; a.:) • IX a 2 aJ +| 
+||a“ 1 a“ 2 a 3 
3 
3 h -j- &• 
s ) 
iAf’Af+XO 
1 2 
2h-\-a\ . h+a 2 
X) 
+ j|Ar + “xx +S3 ) + 1 K'C ta, x + -) 
+ jx+x* ' + X) + K+xX ' **) 
+ |IXa 
A ^ | A fc 
h-\-a i . h-\-a 2 . 
A 2 A 3 ) 
