62 Proceedings of the Eoyal Society of Edinburgh. [Sess. 
the duplicant of the orthogonant | \ of base cr, is equal to 
(Saxni2 - 2o- + Saxnii +1)2-1. (XVII) 
From (XVI) the sum of those of the 4th order 
= Saxni22 - 4(rSaxm2 + ; 
from (XIV) the sum of those of the 3rd order 
= 2Saxm^ . Saxra2 - IcrSaxm^ ; 
from (X) the sum of those of the 2nd order 
= Saxnij2 + 2Saxm2 - 4cr ; 
and by inspection the sum of those of the 1st order 
= 2Saxiiij . 
It is readily verified that the total of these four expressions is as stated. 
(16) There is no corresponding theorem in the case of the duplicant of 
the three-line orthogonant, the total in question then being 
2o-^(Saxm^2 _ 2o-^Saxmj + a) 
+ Saxm^2 + 2Saxni2 - 3o- 
+ 2Saxrn^ , 
which in its simplest form is 
(2(t^ + l)Saxni]2 + (2 + 2o-- - 4cr)Saxm3^ + (2o-t - 3 (t). 
In the case of the duplicant of the orthogonant | \ we have 
2ttj + f^i 
— 4a^y52 2a2/?j (Sf 
= («1 + + 2 I tti^2 I -o-f- jSf 
= (ttj + |32)2 - 2o- + 2o- 
and 
= Saxnij2 . 
2a^ + 2y^2 = 2Saxnq ; 
so that the sum of the coaxial minors of every order is 
(Saxni2 + 1)2 - 1 . 
There is a probability, therefore, that the theorem of the preceding para- 
graph can be extended to all determinants of even order. 
Capetown, S.A., 
September 1914. 
{Issued separately February 24, 1915.) 
