1888 .] Prof. Anglin on Theorems mainly Alternants. 385 
then 
(I)'- 
Taking the first of these expressions and clearing of fractions, (I)' 
becomes 
! Vl I I «0^lC2 I I I i h'~-i 1^-1 I I «0^1«2 I 1 I I I 
+ I Vs I I «0^1«3 I I I I Vs i = 1 Vl I I h«2 I I Vs I I “l^S^S I ^ • 
Now the complementary theorem of this with respect to | | is 
I 1 I ayd-^ \ ^ — dyd^ | a-yd.^ | [ a yd . 2 \ ^ + dyd 2 1 < 2^02 | i ^ 0^3 1 ^ 
■^d^^\a2d^\\ayd^\\ayd2\ (C)', 
and this we have to prove. 
Expanding the squares and arranging the terras, it is easily seen 
that the left-hand side of (C)' becomes 
1 aydyd^ | ^aydyd-yd2dr^ j ayX(^d^ | 
-f d^^^a^d^^ I ayd^ j — a^yd-^^, [ ei-^^ | -f a^d-^ 2 1 a^2 1} 3 
which, since the first two determinants vanish, is by theorem (C) of 
the former paper, equal to 
d^ I ^ 2^3 I 1 ^ 1 ^ 3 1 1 a^2 ! • 
It is thus seen that the new theorem (C)' is derivable from the 
former theorem (C), by the addition of two zeros or vanishing 
expressions. We also observe that, since the expression forming 
the right-hand side of equation (I)' contains no determinants of a 
lower order than the third, its value is unaltered by a double inter- 
change of the letters (that is, by substituting for 5, c any other two 
of the letters a, 6, c), — ^which thus accounts for the two additional 
expressions denoted by the symbol (12). 
Again, with the new notation, the theorem (2) may be enunciated 
thus : — 
If (123) denote any one of the four expressions 
K I \e^d^ I ^ K I ajC2C?3 | ^ 
1 ^0^2^3 1 I 1 1 ! 1 ^0^2^3 i I ^o^l^3 I 1 j * 
K I a^2^?, I ^ ^ I 1 ^ 
I «q&2^3 1 I a^hyd,^ \ [ a^hyd2 \ \ a^2^z I I I 1 ^o^i^'2 L * ' 
where K = | ay)y^ 2 ^i 5 then 
