79 
1890-91.] Dr T. Muir on some unproved Theorems. 
2|(2i&2 C 3^4l‘|^'iA t '2 1/ 3P4l 
E|«i& 2 cA|-|A. 1 /x 2 v 3 <r 4 | 
the difference between any two lying in the second factors only. 
Conversely, if 
Aiai+. . .+A 6 a 6 ^1+. ..+M6^6 V1<»1+. . .+V6 a 6 P1&1+- • •+P6 ct 6 °"1 (X 1+- • •+°'6 ct 6 n«l+* ■ •+T6 f( fi 
A. 1 & 1 +. . .+A6&6 P-1&1+- • -+M6&6 
A]Ci+ . . .+A 6 C 6 p-iCi+ . • .+M6 C 6 
Aldl+. . .+Agdg p.]di+ +^6^6 ! J 
or, as we may write it, if 
f h 
a 2 
% 
a 4 
«5 
a 6 
h 
h 
h 
\ 
\ 
c i 
C 2 
C 3 
C 4 
C 5 
C 6 
•h 
d% 
d 4 
C ? 5 
d 6 
it follows that 
a i 
<h 
a 3 
« 4 
% 
«6 
\ 
h 
&4 
c i 
C 2 
c s 
C 4 
C 5 
C 6 
d. 
d 2 
d^ 
d 4 
^5 
^6 
provided that W^PP'^qI = 0 * For, as we have seen, the fifteen 
given equations are — 
2|^i& 2 CgC? 4 |*|Xi/X 2 V3P4| = 0, | 
S|a^ 2 c 3 ^ 4 !-|X 1 /^ 2 v 3 o- 4 | = 0,1 
— 0} 
where the summation-sign refers to the suffixes, and indicates that 
every possible set of four is to be taken out of 1, 2, 3, 4, 5, 6, and 
that the four indices of the second factor are always the same as 
those of the first; and if we solve for the fifteen unknowns 
|aj& 2 c 3 d 4 j , , ■ • » • we mu st obtain the 
