152 Proceedings of the Royal Society of Edinburgh. [Sess. 
Pi + fi + Pl + yl + yl + % 
a 3@3 
a 5^5 3 
a 2d 3 - a 4T4 
- «575 3 
a 2 d 4 + a 3 y 4 
~~ a 5^5 ’ 
a 2^5 + a 375 
+ a 4 S 5 . 
(14) Let us denote the determinant of the whole of the quadric portion 
by Q and its (r,s) th element by q rs ; also the sum of the squares of the 
elements on one side of the diagonal of the given skew determinant by N, 
and the five primary minors of the quasi -pfaffian of the same determinant by 
F, , F 2 , F 3 , F 4 , F 5 , 
We then find as a curious analogue to the result in 
11 
Qn 9i2 
9.21 922 
915 
925 
951 952 • • • 9 55 
fol N-(Il+B| + p+F» 
3 12 N - F,F 2 
2l5^" ^ 1 ^5 
g-giN-FjFg ? 22 N- 
-(FI+FHF1+F?) .... 
?2 5 n-f 2 f 5 
9 5 2 N ~ F 2 F 5 
g 55 N-(Ff + Fl + I| + FI) 
so that, when the five F’s vanish, the square of Q is elementally a multiple 
of Q, and 
9n ~ 2-N” 9i2 9 i% .... 
*?21 922 ^23 .... 
?31 9-52 933 ~ 2 -^ • • * * 
is an orthogonant. 
In counting the arbitrary parameters involved in the result it is neces- 
sary to remember that the conditions 
f 1 = f 2 =f 3 = f 4 = f 5 = o 
are equivalent to only, four independent equations, it being easily shown 
that the vanishing of any four of the F’s entails the vanishing of the fifth. 
(15) A word must be added in regard to the determinants P and Q, 
which possess a real interest apart altogether from their connection with 
the construction of orthogonants. 
In regard to P, the first point to be noticed is that although it equals 
the square of 
a (3 1 
— a . y e 
- /3 - y .8 
-i -s . 
/ 
