The Substitution Groups of Order 8p. 117 



by dislocation are the dominant ones. In this event the sign 

 of the specific rotation will change as we pass from the un- 

 broken to the broken condition. 



WyroubofFs discovery that in the case of rubidium tartrate, 

 which is optically active in both states, the sense of the 

 rotation is not the same in both, may be mentioned in this 

 connexion*. 



We see from the foregoing that exactly corresponding to 

 the five classes into which substances that possess the power 

 of converting plane-polarized light into circularly-polarized 

 light can be divided, there are five classes of structure dis- 

 tinguishable by characteristic geometrical features. 



It can therefore hardly be doubted that circular polarization 

 is a mechanical effect depending on the relative situation of 

 the ultimate parts of bodies, and that the disappearance of the 

 property and the changes in it observed when the state of a 

 body displaying it alters are also mechanical effects entirely 

 due to changes in geometrical configuration. 



XVII. The Transitive Substitution Groups of Order 8p, 

 p being any Prime Number. By G. A. Miller, Ph.D.\ 



IN a recent paper published in this Journal \ we determine^ 

 all the possible operation groups of order Sp. The present 

 paper is devoted to the more general problem of determining 

 all the possible transitive substitution groups of this order. 

 In solving this problem it is convenient to employ the results 

 of the preceding paper together with the following two 

 theorems. 



Theorem I. — The number of transitive substitution groups 

 which are simply isomorphic to an operation group (G) is equal 

 to the number of different systems of subgroups off (G), such 

 that each system includes all the subgroups that are transformed 

 into each other tohen G is transformed by all the operations that 

 are commutative to it, and none of these systems includes any 

 invariant {self-conjugate) subgroup of G ivith the exception off 

 identity. The substitutions off these simply isomorphic transitive 

 groups can be directly obtained from G, and the degrees of these 

 groups are the quotients obtained by dividing the order off G by 

 the orders off the subgroups in the given systems § . 



* Journ. de Physique, 1894, p. 451. 



t Communicated by the Author. 



\ August 1896, vol. xlii. pp. 195-200. 



§ Gf. Dyck, Mathematische Annalen, vol. xxii. p. 90. 



