622 Notices respecting New Books. 



function. But in the dyadic notation v x f<x=Vv 1 ^(r. Hence 

 the dyadic formula is true only if a is constant or if V-Vip^ 

 vanishes. 



Because of the great importance of the subject, we have given a 

 fairly full description of the essential parts of Professor Gibbs's 

 dyadic theory. It certainly gives us nothing more in the way of a 

 practical working vector analysis than we already possess in the 

 Hamiltonian system. The so-called indeterminate product as 

 such is useless. Cay ley has said that " a product which is not 

 associative has no meaning until the grouping of the factors is 

 determined." To such a category evidently belong the triads and 

 tetrads hinted at ; and the dyad regarded as a product is so far like 

 unto them. Not till it is used as an operator does the dyad take 

 on a determinate meaning ; and then it is found to be nothing more 

 than a bit of the linear vector function, one of the most beautiful 

 of Hamilton's discoveries. The Gibbsian dyad is in fact a kind of 

 lay figure for decorating with notations. 



There is, of course, no fundamental reason why vectors should 

 obey the associative law in products ; but we have only to try to 

 master the meanings of Professor Gibbs's combinations of dots and 

 crosses with the vector operator v> to be convinced that the neglect 

 of the associative principle leads to an increased complexity with 

 absolutely no advantage whatever. The one excuse, it seems to 

 us, for elaborating a vector analysis in rivalry to that developed by 

 Hamilton is that a greatly superior thing is being presented. In 

 the dyadic system of vector analysis we find no evidence of 

 superiority. On the contrary, it is demonstrably more arbitrary, 

 more complicated, and less flexible than the quaternion system. Had 

 the quaternion system been unknown, the other would have been, 

 as a kind of shorthand notation at any rate, a welcome aid in 

 physical research ; but when we bear in mind that Professor 

 Gibbs deliberately set out to construct a system free from the 

 fancied blemish of the quaternion and yet did not scruple to 

 introduce in its stead an indeterminate product which is without 

 any geometric significance whatever, and when we find on careful 

 comparison that 'practically the dyadic system is simply a modifica- 

 tion of quaternion methods, in large measure a mere difference of 

 notation, we can find no satisfactory reason for a man of Professor 

 Gibbs's great powers leaving quaternionic paths to invent new 

 notations, new names for old things, and an indeterminate purely 

 symbolic product to take the place of the determinate real quaternion. 



Whatever views, however, may be formed as to the merits or 

 demerits of the system, there can be only one opiuion as to the 

 zeal, ability, and self-abnegation with which Dr. Wilson lias 

 fulfilled his task. Professor Gibbs is indeed to be congratulated 

 in having a pupil so capable of producing in systematic book-form 

 the subject matter of his lectures. C. G. K. 



