1888.] Prof. Anglin on Theorems mainly Alternants. 
381 
Monday, \Sth June 1888. 
The Hon. Lord MHAKEN, Vice-President, in the Chair. 
1. Exhibition of Photographs. 
The Secretary exhibited M. Amagat’s Photographs of the Crys- 
tallisation of Chloride of Carbon under pressure alone. 
The following Communications were read : — 
2. On the Development and Life-Histories of the Food 
and other Pishes. By Professor W. Carmichael 
MTntosh, F.B.S., and B. B. Prince, Esq., St Andrews 
Marine Laboratory. 
3. On certain Theorems mainly connected with Alter- 
nants (II.). By Professor Anglin, M.RI.A. 
1. In a former paper on this subject,* we have seen that there 
are twm expressions denoted by the symbol (12), and three denoted 
by the symbol (123), in regard to which the theorems 
(23) -(13)-!- (12) = 
I 
I 2 
3 I 
and 
(234)- (134) -F (124) - (123) = 
1 I I 1 1 ^1^2 I 
I I ^ 
1 ^2^3^4 I I ^1^3^4 I I ^1^2^4 i ^'1^2^3 I 
are true. The reason of this is clearly assigned by the geometrical 
interpretation ; since in the former case there are two sets of 
triangles (by means of either of which the area of the triangle ABC 
is obtained), intercepted by the two co-ordinate axes, and in the 
latter there are three sets of tetrahedra (by means of any one of 
which the volume of the tetrahedron PQRS is found), intercepted 
by the three co-ordinate planes. 
The geometrical interpretatiou of results in Cartesian co-ordinates 
further suggests the investigation of similar corresponding results 
by' the use of Tril inear and Quadriplanar co-ordinates ; and since 
Cartesian co-ordinates are really only a particular case of these, 
* Proc. Pmj. Soc. Edin., vol. xiii. p. 823. 
