224 Proceedings of Royal Society of Edinburgh. [sess. 
The second section in like manner opens with an algebraical 
theorem, viz. (p. 13) — 
{^mfe-2/n)}R(2/ c -2/6)} 
+ {x m (z p -z n )}{x a (z c -Z 6 )} 
~ ^n) } ~ %b) } 
= (x m x a )(x p x c ) - (x m x c )(x p x a ) + (x n x a )(x m x c ) 
- (Vc)M + (x p x a )(x n x c ) - (x p x c )(x n x a ) 
+ {x m x b )(x p x a ) - (x m x a )(x p x b ) + (x n x b )(x m x a ) 
- (%n%a)(v b ) + (x p x b )(x n x a ) - (x p x a )(x n x b ) 
+ (y^)M - (x m x b )(x p x c ) + (x n x c )(x m x b ) 
- (x n x b ){x m x c ) + (x p x c )(x n x b ) - (x p x b )(x n x c ), (XXIX. 2) 
where {x m {y p - y n )} and (x m x a ) stand for 
(x m y p x p yf) + {x n y m x m y t ij + (x p y n x n y p ) 
an d x m x a + y m y a + z m z a 
respectively ; and the remainder is occupied with the applications 
of the theorem to geometry and dynamics. Each factor of the left- 
hand side of the identity is evidently a determinant of the third 
order, and the three pairs of lines on the right-hand side are each 
the expansion of a determinant of the same order : so that in the 
notation of the present day the identity may he written 
x m 
2/m 
i 
X, 
X 
y a 
l 
X m 
Zm 
1 
X a 
Z<x, 
1 
x n 
yn 
l 
• 
X 1 
!> 
y & 
l 
+ 
X n 
Z n 
1 
X b 
Z b 
1 
x p 
y P 
l 
X, 
2 
y c 
l 
x p 
*P 
1 
X c 
1 
y m 
Zm 
1 
Va 
1 
(x m x c ) 
{x m Xa) 
1 
+ 
y n 
1 
• 
y b 
1 
= 
(x n 
,X C ) 
(x n x a ) 
1 
Vp 
Zp 
1 
yc 
K 
1 
(*i> 
x c ) 
(x p x a ) 
1 
(x m x a ) (x m x b ) 1 
(x n x a ) (x n x b ) 1 
(x p x a ) (x p x b ) 1 
(x m x b ) (x m x c ) 1 
(x n x b ) (x n x c ) 1 
(x p x b ) (x p x 0 ) 1 
