138 Mr Shaw, On dimensional equations [March 10, 



attributes of an actual concrete quantity than it is justly entitled 

 to. 



We may accept in the first place, as usual, that the complete 

 expression of a physical quantity consists of two parts and may be 

 represented by the symbol q [Q], where q represents the nume- 

 rical part of the expression and [Q] the concrete unit of its own 

 kind which has been selected for the measurement of the quantity. 



The unit [Q] is initially arbitrary for every kind of quantity. 

 There exist however certain quantitative physical laws which 

 really express by means of variation equations relations between 

 the numerical measures of quantities. We may take for instance 

 the following to be the expression of Ohm's law: " The numerical 

 measure of the current in an elongated conductor varies directly 

 as the electromotive force between the ends of the conductor." Or 

 Oersted's discovery may be summed up as follows : " The numerical 

 value of the force upon a magnetic pole placed at the centre of a 

 circular arc of wire conveying a current, varies directly as the 

 strength of the pole, as the length of the wire, as the strength of 

 the current, and inversely as the square of the radius of the arc." 

 A very large number of similar instances might be given. 



We may thus take as the expression of a physical law the 

 general form 



q oc x a y& z y ... 



where q, x, y,z... are the numerical measures of the different quan- 

 tities concerned in the relation. 



We may of course express the variation equation in the form 



q = Jcx a yPzv (1), 



where k is some constant whose value in general alters, if we alter 

 the units in which the different quantities are measured, for by so 

 doing we alter in the inverse ratio the numerical values x, y, z. . . 

 We may adopt one of two courses with respect to the quantity k. 



(1) If all the possible variables have not been accounted for 

 we may regard k as a fresh variable. This has been done in the 

 instance first quoted, viz. in that of Ohm's Law, where k depends on 

 the nature of the conductor. Thus the reciprocal of k in that 

 instance is now generally known as the 'resistance' of the conduc- 

 tor, and we re-state the law thus : " The current in the elongated 

 conductor varies directly as the electromotive force between the 

 ends and inversely as the resistance of the conductor," and the 

 expression of the law becomes 



c== k' -. 

 r 



So that we are still left with an equation of similar form, and 



