4 Iv. F. GWYTIIER, Specification of Stress. Part tV. 

 To obtain these results in the simplest form, I write 



P^Q^R^^^'^-^ (8), 



so that 



•■ „ „ mini - n) , , ?' 



pP-nS7 -P^— V V + 2f/i^^„V-<p 



•■ 3' . 

 nS- U\7'^S=^ 2m;^-r-\7\p (q). 



' oj'cz 



By adding the three equations, of which the first 

 equation in (9) is the type, we get 



v4p»/^-(w + «)vV)} = 0, (10), 



which gives a description of i/r, and which is in agreement 

 with the necessary condition that 



p{P+Q + P)-{'>i + n)\7'{P+Q + P) = 0. 

 It will be noted that xj/^ is described but not defined, it 

 will contain an arbitrary function F such that v'^f^=0. 



Returning to equations (g) and omitting some steps 

 in the reduction, we find as a simplification or solution, 



^ „ m- n n , S^ , 



n ^ dx ■ 



with the analogous equations. In these equations we 

 have with (8) and (10) 



p/o-wV-Zo^O, pSo-n\7-So = 0. . . . (11). 



The correctness of the results in (11) may be verified by 

 substitution for P, or S, in (9). 



The six relations given in (7), (9), and (11) consti- 

 tute the relations between the elements of stress which I 

 proposed to find. In them no direct reference is made 

 to displacement or to strain. 



In the previous Fart of this paper, I extended the 

 forms of expression so as to include cylindrical and 



