4 R. F. GwYTI-iER, Specification of the elements of stress. 



about a vertical axis, in which case the different values of 

 0, , 0., and </)3 are too obvious to require description. 



If the single function should be retained, it would 

 be necessary to correct by terms from the complementary 

 function to which we now proceed. 



TJie Coviplementary Function Solution. 



The right hand side of equations (i) must for this 

 purpose be replaced by zero, and the solution should 

 contain six arbitrary functions. 



If we replace each of the stresses by the general 

 linear expression of second-differential coefficients of a 

 function, we shall have 6 arbitrary constants in the 

 expression for each stress, and therefore 36 such constants 

 altogether. 



Substituting in the stress equations, we should obtain 

 three linear expressions of third-differential coefficients, 

 each of which is to vanish, and we should therefore 

 obtain 30 linear equations of condition between the 

 arbitrary constants, leaving six independent and arbitrary 

 constants. 



As in the ordinary theory of differential equations, 

 each independent arbitrary constant corresponds to an 

 arbitrary function. This plan therefore leads to the 

 complete solution or specification required. 



The labour is made quite slight by the consideration 

 of the stress equations to be solved : for we may conclude ; 

 — {a) that P will contain no differential coefficient in x, 

 Q none in y, and R none in .c ; — {b) that .S" will contain no 

 second-differential coefficient in either r or .:, T none in 

 either z or x, U none in either x or y. 



