494 RICE 



ART, K 



The series therefore involves first powers and squares of x, y, z 

 and product terms such as xy, xyz, x^y, x^y^. Of course the series 

 for </) will in general extend beyond such terms and may indeed 

 be a convergent infinite series. Before proceeding further, 

 it might be well to point out that Vo is just an ordinary factor 

 of temperature expansion (linear), resembling in fact the 

 familiar 1 -\- at oi the text-book of heat. It is necessary to 

 bear in mind that the state of reference is a state at a given 

 original temperature. If the solid is warmed (or cooled) to 

 another temperature without any application of external forces 

 and creation of stress, straining takes place; for an isotropic 

 material it is a uniform expansion. This is an excellent illustra- 

 tion of the necessity of keeping the notions of strain and of 

 stress clearly separated in the mind. Our instinctive notions 

 of pulling, pushing, twisting, bending bodies into different shapes 

 and sizes gives us an unconscious bias towards the idea that 

 stress must invariably accompany strain and vice-versa, whereas 

 change of temperature produces strain (change of size at all 

 events, if not a change of shape which generally accompanies 

 heating of crystalline material) without stresses being created, 

 and if we prevent the strain occurring we have to exert external 

 force on the body with the creation of internal stress, sometimes 

 of relatively enormous value. (We can all recall the experi- 

 ment in our lecture course in elementary physics when the 

 demonstrator fractured the red-hot bar, or the clamps which 

 held it tightly at its ends, by pouring cold water over it.) If 

 therefore we alter the temperature of the (isotropic) body and 

 subject it to external force, the principal elongations with 

 reference to the unstressed state of reference at this tetnperature 

 will be Vi/ro, /'2A0, fs/ro] and ipv, regarded as a function of the 

 temperature and the elongations, can be considered as expanded 

 by Taylor's theorem in the form of a series in the relatively 

 small variables (ri/ro) — 1, (r2/ro) — 1, (rz/ro) — 1. This 

 comes to the same thing as regarding ypv (either its "true" 

 value (f){t, E, F, H) or its approximative value x{t, E, F, H)) 

 expanded as a series in ri — r^, r^ — ro, rz — ro. 



Let <f>o, xo be the values of <^ and x when ro is substituted for 

 each of the quantities n, 7'2, ^3 in E, F, H. Let {d(j>/dr)o, 



