6 REPORT 1861. 



For this purpose we employ the gi-oups 



123 123 123 123 123 123 



2^31 2^31 23^1 231^^ 231^ 



3^12, 3^23, 321=2, 331=2, 31-22, s^ags^ 



■which are all easily proved to be groups. Por example, in the second, 



2=31. 3''123 = P233='= 123. 



3=12'. 2331 =P2'33 = 123. 



We readily formed the grouped groups, 



123456789 123466789 123456789 



231564897 231564897 231564897 



312645978 312645978 312645978 



564789123 645789123 456897123 



645897231 456897231 564978231 



456978312 564978312 645789312 



897123456 978123645 897231456 



978231564 789231456 978312564 



789312645 897312564 789123645, &c. 



There are six groups, each containing thi-ee cube roots of 231564897 and three 

 cube roots of 312645978, that is three 6th roots of 231564897. 



All these are mere groups of nine powers, and are therefore no addition to our 

 knowledge of groups ; but they are formed by the process of evolution, as ffroiiped 

 groujjs of roots, comprising of necessity all powers of those roots, whereas such 

 groups are usually formed by the process of involution. 



Eveiy gToup of powers of a substitution which has two or more cu'cular factors 

 of the same order can be written out either by the process of involution, beginning 

 with a principal substitution next to unity, or by the process of evolution, beginning 

 next to imity with a substitution of a Imcer or of the lowest wde); by means of an 

 auxUiaiy gi-oup. 



For example, the eight cube roots of 214365 are written by the auxiliary groups 



which are all that we can employ, as 1^ = 1, 2^=2, 3' =3, when the elementary 

 groups represented by 1, 2, and 3 are of the 2nd order. 



When the circular factors of the auxiliary group are "of an order prime to that of 

 those of the elementary gi-oup represented by 12 3.. in the auxiliary, all the 

 auxiliaries formed by different systems of exponents give grouped gi'oups equivalent 

 to that given when all the exponents are unities in the auxiliaiy. We have just 

 had proof of this, in the three groups last constnicted. But when the circidar 

 factors of the auxiliary are not prmie to those of the elementaiy gi-oups, we obtaiu 

 by certain systems of exponents grouped groups not equivalent to that given by 

 the auxiliary whose exponents are all imity. 



If, for example, we mean ^X by 1 and ^ by 2, the two auxiliaiies ^v o^l give the 

 groups 



which are not equivalents. 



Tlie Influence of the JRotation of the Earth on the Apparent Path of a Heavy 



Particle. By the Professor Peice, M.A., F.B.S., Oxford. 



The problem of the apparent path of a heavy particle as affected by the diurnal 



rotation of the earth, of course comes within the gi-asp of the general equations of 



relative motion. As these last will be found in treatises on mechanics, where such 



