242 Mr. W. Williams on the Relation of Dimensions 



sions of a solid angle, while in 1=s2ttt, where I is the circum- 

 ference of a circle, it is of the dimensions of a plane angle. 



The significance of it as it occurs in electromagnetism in 

 connexion with radial and circuital fluxes will be considered 

 later. 



The following examples serve to illustrate this method of 

 expressing dimensional formulae. 



23. Radius of Curvature. — Bending per unit length of a 



Fiff. 1. 



curve. Let A B be an element of a curve, and let the tan- 

 gents at A and B intersect at 0, making an angle ^6. The 



AP 



angle "dcf> is ultimately = 7=rp. If A P be taken along the 



normal X at A, and A along the tangent Y, then A B is 

 also ultimately along Y, and may be written "fry. Hence the 



curvature becomes ^— , or dimensionally (XY~ 1 )Y _1 = XY -2 , 



where XY -1 are the dimensions of ~d<f>, and Y those of AB. 

 The radius of curvature at is therefore of the dimensions 

 (Y^" 1 ). 



24. Centrifuqal Force. — Let a particle describe the path 



AB with velocity V, the centrifugal force is F = lw — I. 



Taking instantaneous axes at A, as in 23, this becomes 

 dimensionally 



[F] = M(Y 2 T" 2 ) (XY" 2 ) = MXT- 2 . 



Thus the force is directed along the radius X. 



25. Compressibility. — Hydrostatic pressure is of the dimen- 

 sions MXT~ 2 (YZ) -1 , according to the direction of the plane 

 of reference YZ. Strain is here of no dimensions, being the 

 ratio of two concretes of the same kind. Hence the dimen- 

 sions of compressibility are M -1 X _1 YZT 2 . 



26. Rigidity. — "A simple shear is a homogeneous strain in 

 which all planes parallel to a fixed plane are displaced in the 

 same direction parallel to that plane and therefore through 

 spaces proportional to their distances from that plane" (Kelland 

 and Tait's ' Quaternions,' p. 204). Let the planes of displace- 



