JSotices respecting J^ew Books. 615 



artificial notation, which is simply a commutation of that nsed by 

 Hamilton's brilliant contemporary, O'Brien. On the other hand, 

 if we find that Gibbs's conception of the "indeterminate product'* 

 known as the "dyad" is fundamentally more important than 

 Hamilton's conception of the quaternion, and that this general 

 indeterminate product is analytically more effective than the 

 quaternion, then we cannot but recognise the value of the nota- 

 tion used by Professor Gibbs. When we have occasion to 

 discuss them we shall speak of these rival notations as the dyadic 

 and quaternion notations respectively. 



There is a widespread feeling that vector analysis is a growing 

 necessity for physical investigation ; and yet some of the most 

 strenuous supporters of this view have protested that the vector 

 analysis developed with great fullness and logical rigour in 

 Hamilton's quaternions is not what is wanted. These criticisms 

 find a faint echo in the preface to the present volume. It is said 

 that " Heaviside has set forth the claims of vector analysis as 

 against quaternions, and others have expressed similar views." 

 And again, " As yet, however, no system of vector analysis which 

 makes any claim to completeness has been published." Now in 

 the first place there is no antagonism betw r een quaternions and 

 vector analysis, since the most completely developed system of the 

 latter is to be found in the former. In the second place, although 

 till now no general treatise called " vector analysis " has been 

 in existence, it is simply against the facts to state that no system 

 which makes any claim to completeness has ever been published 

 On the contrary it is a fact which it is pure sophistry to deny 

 that in Hamilton's two great works there is contained a complete 

 system of vector analysis in a highly developed state. 



The main objection urged against the quaternionic system by 

 Professor Gibbs (see 'Nature' vol. 43, pp. 511-12) is that neither 

 the quaternionic product nor the quaternionic quotient of two 

 vectors can claim a prominent or fundamental place among such 

 fundamental geometrical conceptions as the sum of vectors, the 

 vector product of two vectors, or the scalar product of two vectors. 

 This is perhaps largely a matter of opinion ; and yet after a student 

 has formed a clear conception of the vector as a directed quantity 

 is it not reasonable and natural for him to ask, What is involved 

 in the conception of the quotient a//3 as the operator which changes 

 (3 into a? This is Hamilton's method, and leads at once to the 

 conception of the quaternion, and the whole system of vector 

 analysis unfolds itself naturally and consistently. 



Professor Gibbs's method is very different. He first defines as 

 independent and fundamental conceptions the " direct " and 

 "skew" products of two vectors, the direct product being Hamilton's 

 scalar of the product of two vectors with the sign changed, and the 

 skew product being the vector of the same quaternion product. 



Now these well-known quantities are not factorisable products. 

 They may be taken to be, as the quaternion nomenclature clearly 

 shows, parts of a real product which can be factorised, in the sense 



