Notes on the Method of Dimensions. 697 



(III.) Let Q 1? Q 2 , . . . Q/ c be any k of the Q's that are 

 of different and independent kinds, no one being 

 derivable from the others ; and let P. r be any one 

 of the remaining (n — Ic) quantities. 



(IV.) Then equation (1) is reducible to the form 



/(n l5 n 2 , . . . n„_,) = o, ... (2) 



in which the (n — k) quantities II are independent 

 dimensionless products of the form 



U x = Qr.Q|...Qi'.P-r. ... (3) 



In simple cases, the expressions for the IPs may be 

 written down upon inspection; but in any event, the 

 k exponents of each II may be found by subjecting each 

 of the (n—k) equations (3) to the condition that II X shall 

 be of zero dimensions in each of the fundamental units. 



In most instances, the application of the II theorem 

 presents no difficulty, and the mere- algebra is always so 

 simple that whatever doubts may arise relate either to 

 the sufficiency of the list of quantities which appear in the 

 initial equation corresponding to (1), or to the number 

 of fundamental units which should be used for expressing 

 their dimensions. On these two points, questions some- 

 times occur, to which the answers are not obvious at first 

 sight, but which must be answered before we can feel 

 entirely satisfied with dimensional reasoning as a general 

 method. Among these questions are the following : — 



(a) How is it to be known that the initial equation is 



complete, i. e. that the list of quantities is sufficient 

 for the problem in hand ? 



(b) Are dimensional constants to be regarded as " quan- 



tities " ? 



(c) How are universal constants, such as Joule's equivalent 



or the speed of light or the gravitation constant, to 

 be treated ? 



(d) How many fundamental units are needed ? 



In the following notes these questions will be considered 

 more or less in order, although they are so closely related 

 that the answers to them cannot be kept entirely separate. 

 That the answers given are correct will, I think, be 

 admitted, when once they have been exhibited by illus- 

 tration, and no attempt will be made to give general 

 proofs, because such demonstrations would degenerate into 

 mere algebra and present little interest to physicists. 



Phih Mag. S. 6. Vol. 42. No. 251. Nov. 1021. 3 A 



