Note on Double Alternants. 
which (Educ. Times, Ixv. p. 139) 
181 
Sa, - 3 . 
(V.) 
8. Turning now to alternants of the form D„ .„+, let us consider first 
D2.3. By the multiplication-theorem there is obtained 
3a; 
3a^ 
/3x/3a 
3a, 
3a, 
1 
1 /3. /3; /3: 
1 /3. /3; /3; 
1 iC^ x- 
1 y y: 
(a,+/3,)3 (a,+/3,)3 
(a, + /3,)3 (a, + /3,)3 
(a,+-^)3 (a,+y)^ 
(a^ + xy- (a^ + yy 
(j;-/3,)(a?-/3,) (y - i3r)(y - iX) 
x(x - i3,(x - /3,) y{y - i\){y - /3,) i, 
the division of both members of which by 
(y 
gives 
(2/-/y(2/-/5,) 
{X- 
/3.)(^-/3.) 
ttj 3aJ 
3a, 
1 
a^ 3a; 
3a, 
1 
-2/3, 
1 
/3./3. - 
-2/3, 
1 
The four-hne determinant here, however, contains the factor 02 — a,, which 
being removed and a self-evident simplification effected, we have for the 
remaining determinant 
— aia2 . z^a, 
a, 4- a,a, -|- a; 
Mi, 
- 3aia2 
3Sa, 
-2/3, 
/3,/32 -2/3, 1, 
and by performing on this the operations 
Col^Xaiaa, row^-^-aiaa, roWz -f 2ai.r0Wi 
