386 
Proceedings of Royal Society of Edinburgh, [june 18 , 
(234) -(134) + (124) -(123) 
_ I I ^ 
! I I I I 1 1 «0^lC2^3 I 
Taking the first of the expressions denoted by (123), and clear- 
ing of fractions, equation (II)' becomes 
I ^ 0 ^ 1^2 I i Vl^3 I I I I S^l^3^4 I 1 «0^1^2<^4 I I «0^i^2^3 I 1 ^2^3^4 I ^ 
— ... — 1 1 &qC 2^^41 1 bgi^df\ \ af>2^^d^ \ \ af-^cg%^ \ \ajjyi2df[ 1 ^1^2^31^ 
= I ^ 0 ^ 1^2 1 I Vl^3 I 1 ^0^1^4 I 1 ^0^2^3 I ^0^2^4 I I ^0^3^4 I I ^l^2^3^4 I ^• 
How the Complementary theorem of this with respect to 
\\%b^C 2 d^e^\ is 
^2%^4 I ^ 2 % i i *^2^4 I 1 ^3^4 1 I I ^ “ • ' • ~ ^ 1 ^ 2 % 1 *^2^3 I ! ^^1^3 I I ^ 1^2 1 I ^0^4 1^ 
= - I 1 I i I \ \ j | a.^e^ \ \ 1 . . . . (C^)', 
which we now proceed to prove. 
Expanding the cubes and arranging the terms, it may be directly 
shown by the application of equation (B) of the former paper, that 
the left-hand side of (C^)' becomes 
®0^^l‘^2^3^4 I ^2^25 ^3^? ^4^ 1 “ I ^ 2 %> ^3^3? ^4^ I 
+ 3(XQe0-"ej^e2%*^4 I ^1^5 ^2^5 ^4^ I 
- \ «3«4 1 I %«4 ! ! “ 2 % I - a-2hH<^i I «X«3 1 I «1«4 1 I “s^X I 
+ I ^^62 j 1 I I ^ 2^4 1 ^4^^1^2% 1 ^2^3 I 1 ^1^3 ! I ^ 1^2 1} * 
But the determinants in the first three terms vanish, and the 
coefficient of is, by theorem (C^) of the previous paper equal to 
I 1 ! «1^3 1 I ^1^4 I I ^2^3 I I ^2^4 I I ^3^4 I * 
It is thus seen that the new theorem (C^)' is derivable from the 
former theorem (C^) by the addition of three zeros or vanishing 
expressions. 
We also observe that, since the expression forming the right- 
hand side of equation (II)' contains no determinants of a lower 
order than the fourth, its value is unaltered by a triple interchange 
of the letters (that is, by substituting for b, c, d any other three of 
the letters a, c, d )^ — which thus accounts for the three additional 
expressions denoted by the symbol (123). 
In like manner it may be shown that, if (1234) denote any one 
of the five expressions 
