496 PROCEEDINGS OF THE AMERICAN ACADEMY. 



in form with gravitational force, and the time component of WqP 

 with gravitational potential. When the particle is not at rest it is 

 evident that just as in electromagnetics we must add to the scalar 

 potential a vector potential, and to the (corrected) gravitational force 

 another force which by analogy we may call gravito-magnetic. In 

 every other respect, moreover, the two problems must be completely 

 analogous. Thus an accelerated particle must give rise to a singular 

 vector field which we should expect to be associated with the flow of 

 a new form of radiant energy.^* 



Appendix. 



Dyadics. 



61. The dyad or formal product of vectors, introduced in 1844 

 by Grassmann under the name of open product, was given a funda- 

 mental position in vector analysis by Gibbs. Gibbs also developed 

 the idea of the dyadic, or sum of dyads, as the most general type of 

 linear vector operator. The dyadic is useful not only in the treat- 

 ment of the linear vector transformations or strains, but also as a 

 mere formal product (or sum of products) which can later be converted 

 into such determinate products as the outer and inner products of our 

 analysis. We shall outline ver}' briefly the form taken by the theory 

 of dyadics in the vector analysis which we employed. ^^ 



If a, b, c, . . . are 1-vectors, then the product expressed by the mere 

 juxtaposition of a and b, namely, ab is called a dyad. The sum of 

 two or more such dyads is called a dyadic, and any such dyadic in 

 an n-dimensional space can be reduced to the sum of n dyads. As 

 the dyad is in part defined by the assumption of the distributive law, 

 every dyadic in four dimensional space may be expressed as a block 

 of sixteen terms analogous to a matrix. Such an expansion is of great 



64 It should, however, be noted that there is nothing in electromagnetics 

 corresponding to the vector of extended momentum of energy moving with 

 the velocity of light. It is, furthermore, to be noted that while the radiation 

 fields produced by the acceleration of two electrons, whether of the same or 

 opposite sign, due to their interaction, are cumulative, that produced by the 

 acceleration of two material particles, due to their gravitational attraction, 

 must tend to compensate one another. (Cf. the pap?r of D. L. Webster, 

 These Proceedings, 47, 569, 1912.) 



65 For further developments we refer to Gibbs's work as set forth in his 

 Scientific Papers, 2, in the Gibbs-Wilson text on Vector Analysis, and in 

 Wilson's "On the theory of double products and strains in hyperspace," Trans. 

 Conn. Acad., 14, 1. 



