204 Proceedings of the Royal Irish Academy. 



results suppose /? to be variable, and do not introduce tbe reduction 

 /3'^= 1, at all events before differentiation ; Hamilton's results suppose 

 /3 to be constant, and introduce the reduction (3- = 1. 



The expression e'''^'^^ denotes the circular angle b in the plane 

 which has the axis ^. It merely makes definite the ordinary algebraic 

 expression for a circular angle, namely e^'^~^. 



I^ow e^''~^ = cos ^ + J- 1 sin h, 



and corresponding to it 



e^'~P = cosb + J^ sin h /3. 



In Hamilton's notation this is Uq = SUq + VUq. By differentiating 

 the left-hand expression with respect to time, we obtain 



dt dt 



-j-Ki''-f) 



By differentiating the right-hand expression, we obtain 



db ,— db . d/3 



-smb- + J-leosbj^fi+J-lsinb-, 



which cannot be reduced to the result obtained from the left-hand 

 member, excepting under the condition that /8 is constant. The 

 latter result is essentially unsymmetrical, and the reason for it is that 

 the first term is supposed to be independent of /5. As a matter of 

 fact, 6* "^^ = cos b + J- 1 sin b . (3 is not a complete equivalence ; the 

 left-hand expression is a reduced form of the complete equivalence 

 for e* "^^. As the reduction is effected by the principle that /8^ = 1, 

 the differential coefficient obtained fi'om the reduced expression can 

 only be correct when /3 is constant. On the same principle, namely 

 that )8 is constant, 



— d^ o 



= -J-13b^-Jif^' 



and the quaternion analysis makes it so. Hence we have found out 

 the true way of differentiating an integral power of a vector logarithm 

 when both its magnitude and axis are variable. 



