236 Mr. W. Williams on the Relation of Dimensions 



similar manner other such systems may be developed. Thus, 

 take the equation 



f=m|^ +kv, 



ot 

 where F is an impressed force, M the mass of a moving body, 

 v its velocity, and k a coefficient of resistance. Compare this 

 with -sp 



where E is the voltage of a closed circuit, L its self-induc- 

 tion, C the current, and R the resistance. Let us identify 

 E and F dimensionally, and work out the analogy in detail. 

 To do this, we have only to equate the dimensions of E and 

 F, and find what value of /jl satisfies the relation. Then sub- 

 stituting this value of fi in the formulae of the other quantities, 

 we are able to complete a connected dynamical analogy of 

 electromagnetism by starting with voltage as a force. In this 

 case, w T e find that electrification is a displacement ; current, a 

 velocity ; electrical potential, force ; quantity of magnetism, 

 linear momentum; self-induction, inertia, &c. In a precisely 

 similar manner, by starting with the equation 



ot 

 where G is a couple, I a Moment of Inertia, and co an angular 

 velocity, we get: — electrification, a strain; electrical potential, 

 work; electrical force, force; current, angular velocity; quantity 

 of magnetism, angular momentum ; self-induction, moment of 

 inertia, &c. In all cases we are able, by means of the dimen- 

 sional formulae expressed in terms of //, and k, to w r ork in 

 detail any dynamical analogue we may choose to take as a 

 starting-point. Of course all dimensional values of /i } 

 together with the corresponding ones for k, must render 

 electromagnetic dimensions unique. It is only some of these, 

 however, that give rise to formulae capable of dynamical 

 interpretation, and it will be found on examination that the 

 number of such values is small. Now, if electromagnetism 

 is ultimately dynamical, the dimensions of electromagnetic 

 quantities must be of the same kind (ultimately) as those of 

 ordinary dynamical units. Hence, by examining all the 

 possible cases in which the dimensions of \i and k lead 

 in the case of the other electro-magnetic quantities to 

 dimensions of the dynamical order, we may be able to obtain 

 some light as to the nature of fi and k themselves. To show 

 how this may be done, and the kind of results obtained will 

 be the object of the following paper. 



