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NOTE ON DOUBLE ALTEENANTS. 
By Thomas Muie, LL.D., F.E.S. 
(Read October 16, 1912.) 
1. The first form of alternant to which it is desired to direct attention 
is the particular case of 
|(a.+A)^(«a + /3.)^..(a. + /3„)'i, or D„;,, say, 
where 'p = n, the case where — 1 having been already dealt with by 
Zehfuss (Zeitschrift f. Math. u. Pliys. iv. pp. 233-236). The problem is, 
of course, to find the quotient resulting from dividing D„.„ by the dif- 
ference-product of the a's and the difference-product of the /3's — that is to 
say, the quotient 
"2, «,) . ^^(/3„ /3„ /3„), 
or say 
2. It is readily seen that by row-by-row multiplication we have 
Scij 3ai 1 
3a\ 3a, 1 
3o^ 3a, 1 
S/3x/3. 
1 /3, fi\ 
1 /3. ftl 
1 fil 
1 X X- x^ 
(a,+/3,)3 (a, -1-/33)3 (a, 4-/33)3 
(a3 + /3,)3 (a, + /3,)3 (a, -1-/33)3 
(a3 + /303 (a3-h/33)3 (a3-h/33)3 
{a^+xy^ 
{a^ + xy 
(a^ + Xp 
{x-i3,){x-f3,){x-i3,) 
results 
dividin 
^ both si( 
ies of this by (x 
- l3,){x - I3,){x 
al 
3a] 3ai 1 
al 
3a^ 3a2 1 
3a3 3a3 1 
2/3,/3, -2/3, 1 
12 
