2l6 KANSAS UNIVERSITY yUARTERLV. 



The system, however, does not form a group. 



6. The next question is now as to the relation of the invokition 

 of the second kind in space and the system of transformations 

 which leave its cliaracter unaltered. For this purpose equation 

 (^8) niav he written 



U,^ ^ • (12) 



k ,^U,+I 



Designating the invariant elements of (7) b}- m and n. and those 

 of (8) or (12) by p and g, there is 



mn=— pg, 



n— p 



which shows that tlie anharmonic ratio of the four double-elements 

 is — I. Hence, ///r i-r(>ss-ra/i(> of the iiiruiriaiif /^/az/rs of tJir iiiro- 

 liitioii (7) and the traiisJ'oniiatio)i (12) is liarDioiiic. 



If in the expressions for m and n, b and c are of opposite sign, 

 and I be ] ^a^, then m and n are of the same sign and p and g 

 real; when b and c are of the same sign, then m and n are of 

 opposite sign and p and g imaginar}'. 



Equation (g) may be written 



b2+abk 



u-f 



ab — bck 

 ' b2-(-3.bk c 

 ab — bck' b 



b2 + abk b + ak 



u— ki 



-..^_: 



(13) 



