of Physical Quantities to Directions in Space, 241 



16. Angular Acceleration = XY -1 T~ 2 . 



17. Moment of Inertia = MX 2 , MY 2 , MZ 2 . 



Moment of couple _ MXYT" 2 _ M y 2 - 

 Angular acceler. " XY -1 T~ 2 ~" ' 



18. Angular Momentum = Ico = MY 2 (XY _1 )T _1 



-MXYT" 1 , &c. 



19. Energy of Rotation = ila> 2 = MY 2 (XY -1 ) 2 T- 2 



=MX 2 r 2 , &c. 



20. Couple = 1^ = MY 2 (XY" 1 )T" 2 = MXYT" 2 . 



ot 



21. Work done by a couple = G0= (MXYT -2 ) (XY" 1 ) 



=MX 2 r 2 . 



22. Solid angle = ~d£l = — % , where "da is an element 



of area on the surface of a sphere radius r. Taking in- 

 stantaneous axes, X being always along the radius, and Y, Z 

 in the tangent plane at a point, this becomes dimensionally, 



[dn] = (Yzx- 2 ). 



The quantity tt may enter into physical relations in two 

 different ways. It may enter in its purely geometrical or 

 trigonometrical capacity as a definite number of radians (or, 

 what is equivalent, as the ratio of the circumference of a circle 

 to its diameter), and be thus definitely related in a physical 

 sense to the other quantities ; or it may enter in its numerical 

 capacity as part of a numerical coefficient determined by 

 higher abstract analysis. Cases of the former kind only have 

 to be considered in the following paper. In these cases ir 

 enters the relations (ultimately) as a definite number of plane 

 or solid angles, and it therefore consists of two factors, namely, 

 a numerical factor, 3*14' *, and a concrete factor, the unit 

 plane or solid angle. We may therefore speak of the dimen- 

 sions of 7r, meaning thereby the dimensions of the plane or 

 solid angles which it in such cases implies, and the dimensional 

 identity (in the directional sense) of the relations into which 

 it so enters cannot be complete if the concrete factor is 

 neglected. Thus, let A=47jt 2 , where A is the surface of a 

 sphere of radius r. Taking axes as in 22, this becomes dimen- 

 sionally YZ = ( YZX -2 ) X 2 , and the relation is not true if 47r be 

 treated as a pure number unless an area is represented by the 

 square of a vector length, instead of by the product of two 

 different vector lengths as in vector algebra. Similarly, in 

 the relation V=j-7rr s , V being a volume, tt is of the dimen- 



