684 
Proceedings of the Royal Society of Edinburgh. [Sess. 
which by multiplication is expressible as the sum of 24 squares, 12 being 
of the type [la] 2 [a/3] 2 and 12 of the type [la] 2 [/3y] 2 . The second part is 
[ 1 2 3 4 ] 2 + [1235] 2 + [1245] 2 + [1345] 2 , 
%.e. 
{ [12] • [34] - [13] [24] + [14][23] [ * + j [12][35] - [13][25] + [l|[23] } 2 + . . . . 
and therefore is expressible as the sum of 12 squares, all of the type 
[la] 2 [/3y] 2 , together with 12 other terms of the type — 2[la][l/3][ay][/3y]. 
The excess of the first part over the second is thus 
2MW + ZMlfflayl/Jy], 
which is readily found to be equal to 
(13. 32 + 14 . 42 + 15 . 52) 2 + (12. 23 + 14. 43 + 15 . 53) 2 
+ (12 . 24 + 13. 34+ 15 . 54) 2 + (12. 25 + 13. 35 + 14 . 45) 2 , 
or, as it may also be written, 
(»y - 2 ) 2 + (r,r 3 ) 2 + (?y 4 ) 2 + (r x r b f . 
The general theorem here illustrated is 
2[>*1 ! -2W - = Z([i“]M + [i«^]+[iy]-M+ 
-i -l -l -l 
where a, /3, y, . . . on the right is any set of n — 2 integers taken from 
2, 3, ... , n, and r is the remaining integer. Although this is the most 
appropriate form in the present connection, it is better for general purposes 
to transfer 2[i a/3yf to the right-hand side, when it will be seen that we 
-i 
may formulate the result by saying that, A triangular number of elements 
a vl a 
13 
a 
In 
a 
23 
<X-y r 
^ n — 1 , n 
being given, the product of the sum of the squares of those in the first row 
by the sum of the squares of all the others is expressible as a sum of 
J(n— l)(n 2 — 5n + 12) squares, namely, -J(n— l)(n — 2)(n — 3) squares of the 
form (dbiadipy — SiipSiayAaLiydiapy and n — 1 squares of the form (a la a ar -f a^a^ 
+ a ly ai r -f . . . ) 2 . (xxxiv.) 
29. With regard to the cofactor of a n ~ u there is the analogous theorem 
-1 -1 -1 
= 2 | [la][ar , s£] + [l/3][/3rs£] + ... j (xxxv.) 
where a, /3, y, . . . on the right is a set of n — 4 integers taken from 
2, 3, . . . , n, and r, s, t are those remaining. 
