Notices respecting New Books. 621 



In quaternions, the first is \udp-=^\Y da.^u. 

 The second, if regarded from a quaternion standpoint, would be 

 ( dp-uj = ft Ydu v -or. 

 But to find its meaning indyadics, we must realize the expressions by 

 adding .o- or prefixing a. We then get (1) \dpS'&(T=\\YdaVS'aa 

 which is simply a repetition of the first case, or (2) 



\$crdp.TM = uScrda^.Wj 

 an equation which is true if a is constant, or if Y^a vanishes . 

 But generally fedpcm— fiWSc?aV<r+ NSoda^.m. 



The third relation given is ftda.v X^^^dp.m. This in quater- 

 nion notation is f S^jtzr = f \ S^« v^- 



The fourth relation again deals with dyadics. If we consider 

 the dyadic as representing the linear vector function we find the 

 quaternion relation to be 



\Sdp<po = \ (Sc?aV00" 



where v acts both on <f> and a. In the dyadic notation, however, 

 ^ X <j>.cr means V.v J <piO', so that the relation given by Professor Gibbs 

 is 



\Sdp(fxT= ft Sda V 1 l ff 



and is true only when a is constant or when S^v^ffj vanishes. 



The six relations connecting volume and surface integrals, given 

 on page 400, may be discussed in a similar way, and will be found 

 in no respect more general than the quaternion equivalents, and in 

 some cases less general. One example will suffice, namely, 



jTfcv X (f>= \yia x (f>. 



By definition c?aX0 means in quaternion notation —Ydcup. if the 

 operand vector follows, and <p'Yda. if the operand vector precedes. 

 Thus in quaternions 



and ftyVc?«<7=jrfeiyVv<7, 



where v acts on a and on the constituents of the linear vector 



