Notices respecting New Books. 617 



What then is the use of this indeterminate product f By the 

 simple insertion of a dot or a cross we get the scalar or vector 

 product ; and these are stated to be functions of the indeter- 

 minate product in the sense that when it is assigned the others 

 are determined. But how can an indeterminate purely symbolic 

 product be assigned in any true and real sense ? And what is 

 the functional relationship connecting the indeterminate product 

 with the " dot and cross " products? We know that the quantity 

 axfi — a .ft which is Hamilton's Va/3 + Sa/3 is a quantity which is 

 associative in products and which has a definite geometric meaning. 

 It is in fact the quaternion a/3. Gibbs's dyad, written in exactly 

 the same way, has no such useful properties. Out of it the scalar 

 and vector parts of the quaternion product are obtained by the 

 artificial introduction of a dot and a cross ; there is no indication 

 that the properties of the scalar and vector parts are in any way 

 connected with the properties of the dyad, viewed as a product 

 complete in itself. 



There is, however, another use of the dyad, a use which connects 

 it in a very ingenious manner with the linear vector function. 

 The artifice consists in adding to either side of the dyad a vector 

 with a dot between. We thus get the quantities «fl.p, p.afi which 

 are in quaternion notation — aS/3/>, — (S^ap. If we take the sum 

 of several of these operating dyads we get what is called a dyadic, 

 which then becomes a symbol for Hamilton's linear vector function. 

 Thus 



(«i/3, +«A+«A)-f>== — 2aS/3p= — <p P . 



This is undoubtedly a very neat way of representing the trinomial 

 form of the linear vector function, and it has its merits. For 

 example, by merely shifting the operand vector from the one 

 side to the other we pass to the conjugate function. That in 

 itself is, however, of passing moment and seems to have no 

 analytical significance, for writing the dyadic trinomial in the 

 concise form Professor Gibbs falls back upon Hamilton's 

 time-honoured expression and gets absolutely nothing more. The 

 dyadic method leads of course to a special discussion of the proper- 

 ties of the linear vector function, and the usual cubic is deduced. 

 It is then stated on page 321 that " this equation may be called 

 the Hamilton-Cay ley equation. Hamilton showed that a quaternion 

 (sic) satisfied an equal ion analogous to this one and Cayley gave 



the generalization to matrices The analogy between the 



theory of dyadics and the theory of matrices is very close. In 

 fact a dyadic may be regarded as a matrix of the third order, and 

 conversely a matrix of the third order may be looked upon as 

 a dyadic." This sentence contains a mis-statement, and to the 

 ignorant reader would convey a very inadequate idea of the great 

 services rendered by Hamilton. The truth is that Hamilton's 

 linear vector function is the matrix of the third order, and the 

 cubic equation first e-tablished by him, five years at least before 



Mil. Mag. S. 6. Vol. 4. No. 23 Nov. 1902. 2 S 



