of Physical Quantities to Directions in Space. 259 



not be rational and intelligible as a whole, and whose relation 

 to physical reality depends upon the imposed conditions. 



For the purpose of the present paper, let us impose upon 

 the dimensions of jul and k the condition that the indices of 

 the fundamental units in their formulae are not to be higher 

 or lower than those found in the formulae of ordinary dynamical 

 units. This is, of course, a purely arbitrary condition, and 

 simply expresses that the dynamical analogues to be traced 

 out are restricted to those which are of a simple, natural, and 

 intelligible character. All other cases are purposely excluded, 

 not, of course, as a matter of necessity, but as a matter of 

 convenience, for nothing would be gained by the introduction 

 of formulae whose interpretation would be obscure and un- 

 intelligible in our present state of knowledge. Let us there- 

 fore proceed to deduce from dimensional considerations the 

 various analogies between electromagnetism and dynamics, 

 subject to the arbitrary condition imposed above. 



In the case of ordinary dynamical units we notice that 



1. No fractional powers of the fundamental units occur. 



2. The indices of M are never higher than +1. 



3. ,, •>•> X, Y, Z ,. „ „ +2. 



This, therefore, is the range within which //, must be found 

 subject to the above conditions. 



[Of course, if the fundamental units be not L, M, T, but 

 some of the noio derived units, then fractional powers may 

 occur. Thus, if V (volume) be a fundamental unit, the 

 dimensions of length are V', and of area V*, &c. So long, 

 however, as L, M, T, are fundamental units, we cannot ex- 

 pect fractional powers to occur. For length, or space of one 

 dimension, is the simplest conception of space which we can 

 form, while time and mass (not necessarily that of matter, 

 but tangibleness in general) are fundamental conceptions 

 beyond which we cannot go. Now, all dynamical concep- 

 tions are built up ultimately in terms of these three ideas, 

 mass, length, and time, and since the process is synthetical, 

 building up the complex from the simple, it becomes ex- 

 pressed in conformity with the conventions of Algebra by 

 integral powers of L, M, T. The analytical process, that is, 

 splitting up a complex conception into its ultimate consti- 

 tuents (evolution in Algebra), becomes expressed according to 

 the same conventions by fractional powers, e. g. L = (V)^, 

 [Area] = (V)*, &c. But, obviously, if mass, length, and time 

 are to be ultimate physical conceptions, we cannot give 

 interpretations to fractional powers of L, M, and T, because 



