Notices respecting New Books. 619 



It is far otherwise with the dyadic treatment. Professor Gribbs 

 virtually starts with the definitions of what are to him four distinct 

 operators, v operating on a scalar, a.y operating on a vector, v. 

 operating on a vector, and y x operating on a vector. It is noted, 

 however, that "for practical purposes and for remembering formulae 

 it seems by all means advisable to regard 



. d . d _ d 

 dx J dy dz 



as a symbolic vector differentiator." Remembering formulae! 

 Here again it is difficult to imagine a mind finding rest in such 

 an arbitrarily constructed calculus, when already in Hamilton's 

 and Tait's works there lav to hand an effective vector analysis 

 in which y was a real vector differentiator. The collocation yu i«* 

 defined in chapter vii. as meaning a dyad, and then we read : 

 '•The operators v. and vx which were applied to a vector 

 function now become superfluous from a purely analytical stand- 

 point. For they* are nothing more or less than the scalar 

 and vector of the dyadic VW- The anatytical advantages of the 

 introduction of the variable dyadic VW are therefore these. In 

 the first place the operator may be applied to a vector function 

 just as to a scalar function. In the second place the two operators 

 V- and yx are reduced to positions as functions of the dyadic. 

 On the other hand, from the standpoint of physics nothing is to 

 be gained and indeed much is lost if the important interpretations 

 of v-W and V X w as the divergem-e and curl of w be forgotten 

 and their places taken by the analytic idea of the scalar and vector 

 of VW-" With this last statement we are quite in accord if by 

 VW we understand the purely symbolic indeterminate and uninter- 

 preted product. But it is otherwise with the quaternion quantity 

 VW> to which a real meaning can be assigned, v is then a vector 

 operator and the scalar and vector parts of the result of its operation 

 on a vector have analytically and geometrically just those very 

 meanings which make them all-important in physical investigations. 

 To a worker in quaternions these meanings are always in evidence. 

 "Curl" and "Div" are useful descriptively, but they are not 

 analytical working symbols like Sy. and Vy. ■ When the quaternion 

 method is adopted and the true y associative with itself in product 

 combinations is used, everything develops in a perfectly natural 

 manner, and there is no need for the complicated tabulations of 

 the different types of the second order combinations of the dyadic 

 y, so characteristic of the pages of Professor Gibbs's ' Vector 

 Analysis.' 



We have not space to discuss the integral functions Pot, New, Lap, 

 and Max t, which are analytically inverse functions of the true y, 



* This is loose language. V- and vX are not the scalar and vector of VW. 

 t Thev are discussed at some length in a paper on Recent Innovations iii 

 Victor Theory (Proc. R. S. E., xix. pp. 212-237, 1893). 



