Manchester Memoirs, Vol. Ixii, (191 7) No. 1 5 



Briefly, I shall find the results consequent on an infini- 

 tesimal rotation, components w 1 , w 2 , w 3 , of the axes about their 

 own positions upon the elements of stress, and upon the first 

 differential coefficients of an 'arbitrary vector of components 



u, v, w. 



In each case I shall indicate by 12 1 , 12 2 > ^ 3 > ^ e coefficients of 

 w u (i>. z , w 3 , in the resulting expression of change in any of the 

 elements, so that fl n fi 2 , ii ;1 will be differential operators acting 

 on an element. 



In the case of the elements olf stress, the form of these 

 operators depends on the laws of resolution of these elements. 

 In the case of the first differential coefficients of the components 

 of the (vector, Ithe form depends on the laws of differentialtrdn. 

 In one case the argument may be described as mechanical, in 

 the other case as geometrical. 



I shall replace the nine actual differential coefficients by 

 the letters e, /, g ; a, b, c ; '£,, rj, '(, by which we are accustomed 

 to indicate the elements of strain and the components of rotation. 

 They may be regarded as virtual elements of strain, etc. 



For tfie present purpose, the interest lies in the operators 

 and their employment, rather than in the mode of obtaining them. 



In the case of the element of stress. 



\8Q 8M J 8S ST 8c7 * s 8¥ 2 2 8¥ 3 



In the case of the first differential coefficients. 



8 8 \ . „ / , yr\S ,8,7,8,1-8 8 



— ■- — +2 \g -J) — - c— ■ + o — + 4 — - r] —-, 

 \8f 8gJ J 8a 80 8c ' Si, S£ 



and the values of i2., and ;i can be written down by symmetry. 



The similarity of these expressions is well marlced, and 

 would become more so' if we write e' for 2e , f for 2f, g' for 2g. 



According to our hypothesis, the elements of stress are to be 

 functions of the first differential coefficients, and thence, for 

 example, both P and ^j are to be solutions of i2!X = 0, and 

 their general values are to be obtained from the eight independent 

 solutions iof 



de _ df _ dg _ da _ db _ dc _ d£ _ di) _ d'( 

 o a -a 2(g -/) -c b o £ - ?,' 



These general values having been found, values of other 

 elements may be deduced by cyclic interchange. The discrimina- 

 tion between terms in P and in M* x may be made to depend on 

 such relations (as 



a 2 P=-2T, 0,¥i=-¥3, 

 by which also the elements R, S, T, may be found. 



