616 Notices respecting New Books. 



that if AB = C, then A = C/B. But Professor Gribbs takes no 

 account of the possibility of the existence of such a factorisable 

 product. In a footnote on page 67 the student is promised (if 

 he persevere) the definition of the product of two vectors where 

 neither dot nor cross occurs. This is at last given on page 211 , 

 where we read that " the symbolic product formed by the juxta- 

 position of the two vectors ab without the intervention of a dot 

 or a cross is called the indeterminate product." Six pages previ- 

 ous we are told that " an expression a b formed by the juxtaposition 

 of two vectors without the intervention of a clot or a cross is 

 called a dyad." The dyad and indeterminate product are thus by 

 definition one and the same thing. It is called indeterminate because 

 it is neither a scalar nor a vector*. Since each vector involves 

 three numbers the most general conceivable product should involve 

 six. The dyad, however, involves only five, since the assumption 

 is made that the product of the tensors of the constituent 

 vectors only is involved. Given the dyad, the directions of both 

 vectors are fully determined, so that equality between two dyads 

 means that the constituent vectors are, in regard to direction, the 

 same in both. It is difficult to see what service such a restricted 

 product is to be in any system of vector algebra, nor do we gain 

 any enlightenment from subsequent parts of the book. In fact this 

 product with its five disposable numbers is indeterminate, does not 

 appear to be factorisable, and has no attachable meaning until it 

 ceases to be itself by being defined anew as an operator in combi- 

 nation with other vectors. The quaternion involves four numbers, 

 and having therefore what might be called from analogy two 

 degrees of freedom is fitted to play an important role in vector 

 combinations. It has pleased Professor Gribbs to introduce the 

 one condition that scalar multiplication in vector products is to* 

 be associative. Thereby he gets a purely symbolic indeterminate 

 and uninterpreted product. It pleased Hamilton to introduce 

 the condition that vector multiplication in products of three or 

 more vectors was to be associative. Thereby he obtained a real 

 determinate and fully interpreted product caJled the quaternion. 

 It is very difficult to understand the argument that the real 

 quaternion should be disallowed in a vector analysis, but the dyad 

 as a symbolic indeterminate product welcomed with open arms. 

 There are hints throughout the book regarding higher indeterminate 

 products of vectors called triads, tetrads, etc. ; but the theory of 

 these is not given, there being, we are told, no real need for 

 them in physical applications. As a matter of strict logic, there is 

 no real theory given of the dyad as an indeterminate product. 



* Dr. Heaviside demands that vectors be treated " vectorially," and that a 

 vector analysis be purged of products which are non-vectorial. This is his 

 reason for condemning the quaternion. It is curious that he has expressed 

 intense admiration of the dyad or indeterminate product, which its inventor tells 

 us explicitly is no vector but purely symbolic, and acquires a determinate 

 physical meaning only when used as an operator. Could anything so hope- 

 lessly " un vectorial" be said of the quaternion ? 





