1888 - 89 .] Dr T. Muir on the Theory of Determinants. 217 
occurs in connection with the latter save that use is made of the 
identity, 
ffy" P'y _ a( y e " _ yy) + _ J c ") + a "{bc' -Vc) , 
a 
where ft = la' - 1' a , /3" = l' a" - Ta f , 
y = cal - c'a , y" = c'a!' - c"a ' . 
This identity, it may he remembered, we have noted under Lagrange 
as an elementary case of the theorem afterwards well known regard- 
ing a minor of the adjugate determinant. Strange to say, it makes 
only its second appearance here fifty-six years afterwards. In the 
interim, too, no other special case of the theorem seems to have 
been established. 
The third is that if P, P', P", P'", be four points in space, given 
by the equations, 
q P =a A + 
q' P' = a' A + 
q" P" = a" A + 
q"?"' = a'"A + V"B 
c C + d D , 
c' C + d' D, 
+ c" C + d" D , 
+ c'"C + d'"~D ; 
b B + 
V B + 
b"B 
then for the bulk of the tetrahedron P P' P" P'", we have 
where 
PFF'F" A + A' + A" 
ABC D “ qii'4" 5 
A = 0'(/3 "y'"-ry"), 
and 
P =a' b -a V , 
P" = a" V - alb" , 
P'" = a"'b"-a"V " , 
A' = S''(/3"y - ffy") , 
y —a'c -a c' , 
y" —al'd - a' c" , 
y =a c - a c , 
A" = 3 "W-/3V), 
d' — a! d -ad' , 
I d" = a" dl — a' d" , 
d" = a'"d" - al'd !" . 
The transformation of A + A' -f A" into the form 
a'a"\ab'c"d'"\ 
— a transformation all-important for Minding’s purpose — is not 
made : but in the remark, 
