618 Notices respecting New Books. 



Cayley, is the equation satisfied by the matrix of the third order 

 and by the dyadic viewed as a functional operator. There is in 

 fact identity — no mere analogy. There is, on the other hand, no 

 evidence that the "indeterminate product" with its five involved 

 numbers, or the dyadic trinomial with its fifteen numbers, satisfies 

 an equation of this kind at all. As soon as the dyadic is regarded 

 as an operator of the kind defined, it becomes Hamilton's linear 

 vector function, and of course satisfies the same cubic equation. In 

 fact if, as seems 1o be the case practically throughout the book, the 

 d\adic exists only as an operator, then the dyad should strictly 

 speaking be written a/3, or .a/3, just as in working in quaternions 

 we may put the linear vector function in the symbolic form 

 cijS/3,. +a^ 2 . +n 3 S/5 3 .. meaning that the operand is to follow. 



Professor Gribbs also defines the " skew" products of a dyad into 

 a vector. They are 



(u/3)Xy = «G3Xy), «X(3y) = (aX/3)y 



That is, the original indeterminate product is split up, and one of 

 the members is joined with the new operand to form a new vector, 

 which with the other member forms a new indeterminate product. 

 This is then to be used as a dyad operator. To find what the 

 relation of this new dyad is to the original dyad, we must let it 

 act upon a vector. Extending to dyadics we find the quaternion 

 equivalents to be as follows : — 



(a,/3, + a v /3 2 + . . . )Xf>.«7=-</AV, 

 <7.(« l /3 1 + a 2 /3 2 + . . . )X/0= Yfxp'tT, 



f>X(<« 1 ft+ ).a =~Yp(J if T, 



a.pX(a^ x -\- ) =0'V /Uff , 



where <p is 2nS/3., and </>' is the conjugate 2/3So. Hence these 

 " skew " products are equivalent to the quaternion operators <p\ T p. y 

 V/></>., which in the quaternionic treatment come naturally to the 

 front when needed without the necessity for new definitions. 

 This constant appeal at every turn to new definitions is certainly 

 not a pleasing feature of Professor Gibbs's method. 



The section on Double Multiplication starts as usual from arbi- 

 trary definitions, and seems to us to make serious demands upon 

 the memorizing power of the student. Its value is not very 

 apparent ; for its chief if not only use in the treatise seems to be 

 to arrive at the inversion of the linear vector function. Passing 

 it over meanwhile, we shall confine our remaining remarks to a 

 description of the dyadic treatment of what corresponds to Hamil- 

 ton's beautiful operator v« lu quaternion vector analysis, v is a 

 differential vector operator of the form id l -\-jd 2 -\-kd 3 where 

 d 1 d 2 d z are differentiations in the directions of the mutually 

 perpendicular unit vectors ij k. Prom this single definition (or 

 from any other equivalent to it) the properties of v evolve themselves 

 naturally along the lines of the calculus. 



