476 ' PROFESSOR C. NIVEN ON THE 
the equation of equilibrium, and expressions tor the elastic forces. It is of 
course to be noticed that the expression for the work due to the potential 
energy of strain must be first integrated by parts throughout the volume of the 
solid under consideration. This volume may be any portion whatever of the 
whole solid, or the whole body itself. 
Determination of Stresses and Equations of Equilibrium. 
§ 4. On equating separately to zero the co-efficients of Su... taken over 
the three separate boundary integrals, we obtain nine equations to find the six 
stresses. But this difficulty is explained by the symmetry of the expressions 
for the tangential stresses which indicate that they may be derived from different 
sets of equations by different routes. 
The general type of these fundamental boundary conditions is given by 
dw aw aw 
Se hee Sa Va of Senn | = doe Cs + Glspy M2 + se Us 
+ ae 
aw aw aw 
Se Ue + Sica = S.,- W, = Boye + sy S= ds,, ° “8 
From these we may derive either the stresses in terms of the rate of change 
of W or conversely. The former are given by the general formule— 

aW aw aW 
See V = Ge: + Ge, = ayes +2 GU tills ] 
(7). 
dW dw dw 
See V — Sing CVU + sy 22 + ove + ds, (ty. + Uy) \ 
zy 
These results are manifestly capable of expression in a symbolical form, 
which can be best explained’ by studying any one form and comparing it with 
its more complete development. The form is 
dw 
Sage VS ae (Oe + Ong 4 One) -\Ore tp On, fF Caal e (a), 
where a= 07,0 = 0 w= 6. 
A similar form exists for the expression of i in terms of the stress. 
It is 
dn, V = 8. (Di. + Di, + Di.).(Di.+Di,+Di.) - ©), 
where U = D’, V =D’, W = D’, and the suffixes 7,7 mean 1, 2, 3 according 
asi, pp Mean wv, 7, 2. 
The typical form of the equation of motion or equilibrium is 


Gs  GNVs sa ee ONY, ad aw ASX 
Pb r + dx * ds, + dy * dsy a dz° i) (81,2 rir 82, y oh 63,2)” = Po di (y). 


