388 Proceedings of Royal Society of Edinhurgli. [june 18 , 
second series of theorems become identical with the first series in 
every respect. 
Further, the fact that the third expression for (12) and the 
fourth expression for (123) become infinite by this substitution, is 
verified geometrically ; since the results in the first case correspond 
to Cartesian co-ordinates, and in flie second case to Trilinear and 
Quadriplanar co-ordinates, and in order to deduce the Cartesian 
system from the other system the third side of the triangle of 
reference and the fourth plane of the tetrahedron of reference move 
off to infinity; and thus the triangles intercepted by the former 
and the tetrahedra by the latter become infinite. 
5. Although the foregoing theorems are of a more general 
character than the corresponding ones of the previous paper, and so 
may be regarded as extensions of them, yet they are also in a sense 
special cases of these theorems. That this is the case may be 
gathered from the following considerations : — 
A theorem stated in the former paper is to the effect that, if 
any six quantities whatever, then we have the identity 
Uj ^2 
I ^ 2 ^ 3 1 1 1 1 ^1^2 1 ^ I af>^ I - | | -f apxff | ! • (C) . 
Taking the special case of this where the six quantities are 
di d-i c/3 
1 ^0^1 1 ’ ! 1 1 1 %^3 1 > 
the identity becomes 
df I afi.^ 1 1 1 1 cc^c?2 1 = dglpl.^ | a^d^ \ | a^^d-^ \ ^ 
- d^plfl.^ I cqc/3 1 1 afl .2 i ^ + d^d-pl^ I cCjC/2 1 1 %d^ ] ^ , 
which is the identity (C'), the complementary of which is the new 
theorem (I)'. Thus the old result (I) is the complementary of (C), 
and the new result (I)' is the complementary of a special case 
of (C). 
In like manner it may be shown that the other new theorems 
(II)', (III)', &c., may be regarded as special cases of the corre- 
sponding old theorems (II), (III), &c., respectively. 
6. Leaving this, we shall now return to the general theorems (I), 
(II), &c., of the previous paper, and show how by specialisation 
they give rise to theorems regarding Alternants. 
