THE GEOMETRY OF MOTION 20 







represented by a sphere, and the change in size will 

 depend upon the change in diameter. The ratio of the 

 extension to the original length of the diameter may be 

 taken as a proper basis for the measurement of the strain. 

 Such a ratio is termed a stretch, and it may be shown 

 that for a small increase of size the ratio of the increase 

 of volume to the original volume is very nearly three 

 times the stretch of the diameter.^ This ratio is termed 

 the dilatation, and is a proper measure of the change in 

 size. Now it is clear that in order to measure this change 

 of size, we require to measure the diameters in the two 

 conditions of the body. But a diameter, although in the 

 conceptual body definite enough as a straight line termin- 



Fig. s. 

 ated by two points, is, in this accurate sense of the word, 

 a meaningless term when we are dealing with a perceptual 

 body. If the body has no continuous boundary, but, 

 according to the physicist, is a mass of discrete atoms 

 (Fig. 5), none of which we can individually feel, and the 

 mutual distances of which we cannot measure, it is clear 

 that the only diameter we can be talking about is that of 

 a conceptual sphere by which we have replaced the per- 

 ceptual ball. 



1 The volumes of bodies of similar shape are as the cubes of corresponding 

 lengths. Hence if V and V be the old and new volumes, d and d' the old 

 and new lengths, VlF=d'^/d'^, but if s be the stretch {d'-d)/d=s, or 

 d' = d{i +s). A little elementary algebra gives us for the dilatation 5 : — 

 y- I' d'^-d'^ 

 5 = — y^ = -^3-" = (i+^)^- I = 3->" + 3-f " + ■>•' = 3-s'> nearly, 



if s, as in most practical cases, be very small. For example, in metal 

 ■^ = ToVo would be a rather large value ; but taking 5 = 3^-, we should only 

 be neglecting about x-sVo" of the value of o. 



