312 MILLER — ON THE QUATEENION GROUP. [Oct. 7, 



Videns piscator a militibus se comprehendi putavit occidi. Ingressus 

 Appollonius jussit eum adduci ad se et ait: "hie est paranymphus 

 meus, qui mihi post naufragium opem dedit et ad civitatem venire 

 ostendit." Et dicit ei : "ego sum Tyrius Appollonius." Et jussit sibi 

 dan CC sistercias argenti, servos et ancillas, et fecit eum comiteni 

 suum, quamdiu vixit. Elamitus vero, qui ei de Antiocho nunciavit, 

 procidens ad pedes Appollonii .... et ait: " domine, memor esto 

 Elamiti servi tui !" Appollonius apprehensa manu eius erexit eum fecitque 

 eum divitem et ordinavit comitem. Hiis expletis genuit Appollonius 

 fi-lium de conjuge sua, quem in loco avi sui Altistratis constituit regem. 

 Vixit vero Appollonius cum conjuge sua annos LXXIV et tenuit regnum 

 Antioche et Tyri et Tyrenensium quiete ac feliciter. Casus suos ipse 

 descripsit, ipse duo volumina perfecit, unum in templo Ephesiorum, 

 alterum in sua bibliotheca coUocavit. Et defunctus est et perrexit ad 

 vitam eternam, ad quam vitam nos perducat, qui sine tine vivit et regnat 

 Amen. 



ON THE QUATERNION GROUP. 



BY G. A. MILLER, PH.D. 



{Read Octoher 7, ISDS.) 



Although the quaternion group {Q) has received some attention/ 

 yet many of the properties of this important group remain to be 

 investigated. It is the object of this paper to enter upon the study 

 of some of these group properties after stating the known principles 

 which underlie the investigations that follow. We shall also deter- 

 mine the different ways in which Q may be represented as a substi- 

 tution group. 



It is well known that every group of a finite order may be repre- 

 sented as a regular substitution group and that any two regular 

 substitution groups which are simply isomorphic are also conjugate. 



A complete list of the regular substitution groups of order g 

 must therefore include every possible group of this order and no 

 group can occur twice in such a list. In following Prof. Cayley's 



^ Dedekind, Maiheniatische Annalen, 1S97, ^^^' xlviii, pp. 549-552. 



