244 Prof. C. Nivcn on the Theory of an 



solid may be represented by supposing the elements of this 

 strain and rotation to vary continuously from point to point. 

 If, however, the solid be not perfectly homogeneous in detail, 

 though it may be so on a large scale, it will be necessary to 

 take the element-volume great enough to eliminate the effect 

 of the irregular variation of the different particles of the solid 

 within the element itself. In order to understand how this 

 may be represented, let us first suppose that A, B, C, D, E, F, 

 xs i, «r 2 , vr 3 vary regularly and continuously within the element; 

 then, if these symbols refer to some definite point ¥(x,y, z) in 

 its interior, say its centre of gravity, the value of ■cr 1 at some 

 other point P x (x + h, y + k, z + l) will be denoted by 



7 dtsr, z dvr-i 7 dtffx 



1*1 + h — 7 h k -z h I —, — 



dx ay dz 



But even if we admit that these quantities may vary gradually, 

 though not necessarily regularly, we may suppose that the 

 value of mi at P x differs from its value at P by linear func- 

 tions of the differential coefficients of itself and of the remain- 

 ing elements of the distortion, and that the coefficients are 

 quantities of the order of the magnitude (ttt) which measures 

 the extent through which the coarseness of structure is sensible 

 — that is to say, the molecular distance. Thus the circum- 

 stances of the strain of the element-volume will require for 

 their complete specification the nine quantities above men- 

 tioned, and also their twenty-seven differential coefficients with 

 regard to x, y, z. 



3. Now the known cogrediency of «j 1? ot 2 , vr z , with the dis- 

 placements u, v, w, enables us to analyze their differential 

 coefficients into the two groups : — 



0^1 2 ^2 c)dix z disT Z dvT 2 d^ dtTz d^r 2 dvr lm 



dx ' dy ' dz dy dz dz dx dx dii 



d'ur^ d^2 dvTi d^ s d^r 2 dtiTi 



dy dy dz dx dx dy 



The first group are cogredient with strains, and will be denoted 

 by a, b, c, d, e,f; they represent three uniform twists of the 

 element about three rectangular axes. The second group are 

 cogredient with displacements, and will be denoted by 2^\, 

 2'5J- / 2 , 2'57 / 3 . With respect to the differential coefficients of A, 

 B, ... F, I have not attempted to analyze them further; but 

 this is of no consequence, as they will give rise to terms in 

 our equations which will be subjected in the sequel to a special 

 treatment. It may be useful to make here the remark that 

 the theory just given is equivalent to supposing that, instead 



