40 2 Leigh Page, 



may change in direction from point to point, but the meaning of 

 these quantities at any one point is perfectly definite. While 

 Gibbs' vector methods will be followed, it is necessary to make 

 a slight change in the notation generally used in order to avoid 

 ambiguity in the expression for the scalar product of a vector 

 with a polyadic of order higher than the second. For if 

 H n = X ^' o k kok 



a/3y 



the usual notation provides no way of expressing the dot product 

 of a vector with the middle vectors of this triadic. Consequently 

 we shall adopt the convention that the dot product is always be- 

 tween those vectors whose suffices are the same. Thus the dot 

 product of a vector P with the middle vectors in the triadic H is 

 represented by 



With this convention the order in which vectors and polyadics 

 appear is immaterial ; in fact the dot itself is superfluous, for 

 whenever the same suffix appears twice dot multiplication is 

 indicated. However, the dot will be retained as a reminder of 

 the analogy of the notation employed with Gibbs' three dimen- 

 sional analysis. Obviously the suffices employed in dot multi- 

 plication are dummies, i. e. 



and 



^/3 '^a^-y 



H 



aey 



mean exactly the same thing. Hence we are at liberty to replace 

 a repeated suffix by any other letter which does not appear as a 

 suffix in the same term. An additional advantage of the proposed 

 notation lies in the fact that the number of letters in the suffix 

 shows at a glance whether the quantity under consideration is a 

 vector, dyadic, triadic, etc. 

 Now consider the dyadics 



K ^P"'^ k ', (33) 



Do ^^'"'k ' kp. (34) 



P v/3 QxQ y p 



