QUATERNIONS IN THE ALGEBRA OF VECTORS. 159 



we adopt is a matter of minor consequence. In order to keep within 

 the resources of an ordinary printing office, I have used a dot and a 

 cross, which are already associated with multiplication, but are not 

 needed for ordinary multiplication, which is best denoted by the 

 simple juxtaposition of the factors. I have no especial predilection 

 for these particular signs. The use of the dot is indeed liable to the 

 objection that it interferes with its use as a separatrix, or instead of a 

 parenthesis. 



If, then, I have written a./3 and aX/3 for what is expressed in 

 quaternions by Sa/3 and Va/3, and in like manner V.o> and Vxw 

 for SVo> and VVo> in quaternions, it is because the natural develop- 

 ment of a vector analysis seemed to lead logically to some such 

 notations. But I think that I can show that these notations have 

 some substantial advantages over the quaternionic in point of con- 

 venience. 



Any linear vector function of a variable vector p may be expressed 

 in the form 



where 3? 



or in quaternions 



- aS\p - pSjuLp - ySi/p = - (a 

 where = 



If we take the scalar product of the vector 3>.p, and another vector 

 <r, we obtain the scalar quantity 



or in quaternions Sa-(j)p = So-(aSX + /3S/UL -f ySv)p. 



This is a function of a- and of p, and it is exactly the same kind of 

 function of or that it is of p, a symmetry which is not so clearly 

 exhibited in the quaternionic notation as in the other. Moreover, we 

 can write <r.& for o-.(aX + /5/x-f yi/). This represents a vector which 

 is a function of <r, viz., the function conjugate to $.0-; and cr.&.p 

 may be regarded as the product of this vector and p. This is not 

 so clearly indicated in the quaternionic notation, where it would be 

 straining things a little to call So-0 a vector. 



The combinations aX, /3yu, etc., used above, are distributive with 

 regard to each of the two vectors, and may be regarded as a kind of 

 product. If we wish to express everything in terms of i, j, and k, 3> 

 will appear as a sum of ii, ij, ik, ji, jj, jk, ki, kj, kk, each with a 

 numerical coefficient. These nine coefficients may be arranged in a 

 square, and constitute a matrix; and the study of the properties of 

 expressions like < is identical with the study of ternary matrices. 

 This expression of the matrix as a sum of products (which may be 



