620 Notices respecting New Books. 



and the necessity for which seems to arise mainly from the neglect 

 of the associative principle in product combinations of v* This 

 particular section, however, is very instructive reading, quite apart 

 from the dyadic system of vector analysis. 



The collocation \<p is called a triadie, but its properties (if there 

 be any) are not discussed. The dyadic v X § and the quantity ^.<f> 

 are, however, defined. The former is the quaternion operator 

 "V T V0. ; hence when realized it means not the more general 

 quantity V.vvw, but the particular case V.v^w, where the suffixes 

 mean that v acts on the constituents of q> and not on -m. In 

 this respect the dyadic notation is not so general as the quaternion. 

 The quantity v«4> corresponds to what McAulay writes in the form 

 %V V meaning 



d d d 



ij 7c being the usual rectangular system and the constituents of <f> 

 varying with position. If <f> is written in the form 



<pm = a$im + fiSjm + ySJcm 

 ., da dQ dy 



then #iVl= _ ______ 



So far there does not seem to be the least advantage in the 

 " dyadic " over the " quaternion." It leads to nothing more, gives 

 no greater generality, and is occasionally indeed less general. 



Passing over various examples which a quaternionist would not 

 need to tabulate as they are perfectly simple transformations in 

 quaternions, we come to the equations among line, surface, and 

 volume integrals. Tait has practically given these ; but we owe to 

 McAulay the completely general form which includes all. He 

 shows that if Q be any linear function of a vector 



jQdp=jjQ(Vrfav), JJQcZ«=$JQv^ 



where da represents the vector area element of the surface, bounded 

 in the first case by the curve (p), and bounding in the second case 

 the volume v, and where it is understood that v acts on all the 

 varying factors in the expression. 



Professor Gribbs gives four relations connecting line and surface 

 integrals, namely, 



\\ da X V u = \ dp l h \\ da X \iz = f„i . __•, 



^ tfo.V X -ui = jc/jo.-ar, (pa. V X <p = fo/o.0, 



where u is a scalar function, _• a vector, and <f> a. linear vector 

 function. 





