370 
Proceedings of Royal Society of Edinburgh. [sess. 
be expressed in one and only one way as a sum or difference of two 
other such products. 
(9) We are thus prepared to learn that if we take either of 
the two products whose sum or difference has been obtained in 
this way as an equivalent for a given product of the same kind, 
and apply our theorem as before, we shall merely get another 
repetition of the previous result. Thus 
I #]&2 C 3 I’l I — I a i^2 C S [\ C l92 h 3 I d- I ^1^2 C 3 H c l a 29s I 
i.e. 
23 
or 
+ 
59, 
and 
^1^2 r 3 1 
■| c-yg^a^ | 
= j gf> gCg |*| eft ^a^ 
i + 
! a i^2 C 3 I'l C 1^2^3 1 ’ 
i.e. 
-59 
= -23 
+ 
or. 
It follows 
therefore 
that since there 
are 
ninety products and 
each can only occur once in an identity along with two others, 
the number of such identities is thirty. Probably the best 
arrangement of the thirty is that which brings into juxtaposition 
those that form a triad, and places opposite to each other those 
that are complementary. The result of this is : — 
or 
- 
23 
+ 59 
= 
0 
= 
O'l 
- 
2'3' 
+ 
5'9' 
02' 
- 
31 
+ 67 
= 
0 
- 
0'2 
- 
3'1' 
+ 
6'7' 
03' 
- 
12 
00 
+ 
= 
0 
= 
0'3 
- 
1'2' 
+ 
4'8' 
or 
_ 
56 
+ 38 
= 
0 
= 
0'4 
— 
5'6' 
+ 
3'8' 
05' 
- 
64 
+ 19 
= 
0 
0'5 
- 
6'4' 
+ 
1'9' 
06' 
- 
45 
+ 27 
= 
0 
= 
0'6 
- 
4'5' 
+ 
2'7' 
07' 
_ 
89 
+ 26 
= 
0 
= 
0'7 
- 
8'9' 
+ 
2'6' 
08' 
- 
97 
+ 34 
= 
0 
= 
0'8 
- 
9'7' 
+ 
3'4' 
09' 
- 
78 
+ 15 
= 
0 
0'9 
- 
7'8' 
+ 
1'5' 
ir 
+ 
82' 
-f 69' 
= 
0 
1'4 
+ 
8'2 
+ 
6'9 
25' 
+ 
93' 
+ 47' 
= 
0 
- 
2'5 
+ 
9'3 
+ 
4'7 
36' 
+ 
71' 
+ 58' 
= 
0 
3'6 
+ 
7'1 
+ 
5'8 
16' 
*r 
49' 
+ 73' 
= 
0 
- : 
1'6 
+ 
4'9 
+ 
7'3 
24' 
+ 
57' 
+ 81' 
= 
0 
= 
2'4 
+ 
5'7 
-f 
8'1 
35' 
+ 
68' 
+ 92' 
:= 
0 
= 
3'5 
+ 
6'8 
+ 
9'2 . 
