156 Proceedings of the Royal Society of Edinburgh. [Sess. 
edition contained (pp. 33-37) as a part of § vi. on “Inverse Systems” is 
placed in the second edition under “ Compound Determinants ” (pp. 368-372). 
Sylvester’s mode of reaching Cayley’s determinants connected with the 
mutual distances of points is given under “ Multiplication ” (pp. 250-253) ; 
and the chapter or “section” (§ iv.) on “Homogeneous Functions,” which 
of course has to deal with quadrics, goes so far as to assign the name 
determinant of a quadratic form to any determinant possessed of axi- 
symmetry (see pp. 328, 336). 
The most interesting matter, however, is found in the last section of 
all, § 11, the contents of which are miscellaneous. There, on pages 
376-380, the determinants 
1 a Y a 2 
. 1 \ b 2 
a Y \ 1 
a 2 b 2 1 
a 2 b 2 c 2 . 1 
a s b 3 c 3 . . 1 , 
b i b 2 
are considered, hut, to one’s regret, only with reference to the case 
where \a 1 b 2 c 3 \ is an orthogonant. The first of the two determinants is 
given in the form 
\-af- b 1 2 - a 2 2 - b 2 + {af 2 - a 2 bf 2 ; 
and similar non-determinant forms are given for the whole of the thirty- 
six primary minors and for the first fifteen of the secondary minors of the 
second determinant. Thus, the primary minors which are the cofactors of 
the elements in the places (1 , 1), (1 , 3), (1 , 5) are 
l-b 2 - b 2 - b 2 - c 2 -c 2 - c 2 + (b 2 c 3 - b 3 c 2 ) 2 + (b 3 c Y - b^) 2 + {\c 2 - h fa ) 2 , 
K c i % c s)(l — b i“ — b 2 — 6 g 2 ) + (t\c^ + b 2 c 2 + b 3 c^)(a l b 1 + af) 2 + , 
I ' I b\C 3 | - 0^(1 - b 2 - b 2 2 - b 3 2 - cf - 2 2 ~ c 3 2 ) - bfa^ + a 2 b 2 + a 3 b 3 ) 
- + a 2 c 2 + a 3 c 3 ) , 
all the others being similar in form to one or other of these three ; and in 
like manner the secondary minors are exemplified by 
1 - r % ~ r 2 _ r 2 
i Oj <-'2 j 
~( b i c i + b 2 c 2 + b 3 c 3 ), 
~ bf\ — — c 2 — cf) + C\(b\ G \ + b 2 c 2 + b 3 c 3 ) , 
a i( b i c i + b 2 c 2 + b 3 c 3 ) - bfc^ + c 2 a 2 + c 3 a 3 ) , 
-Ci | af) 2 c 3 | + | a 2 b 3 | . 
