
STRESSES DUE TO COMPOUND STRAINS. 479 
The equations of motion of the solid, as has been remarked by St VENANT, 
assume a very simple form, even in the case where (S°) are not constant 
throughout the substance. If they were produced solely by external force act- 
ing on the surface of the body, the new equations have for their type 
Ged “a Vie esd. do! a? a? 
pX’+(ae- ae taq- det aaa dao ee (S.apt... +28!, 757 — pap )u=0 (13). 
Tf the strains (S°) were produced under the action of forces X,, Yo, Z), we 
must add to the left hand member of this equation the expression 
du du 
§ 9. Resolution of Strains.—In the foregoing results nothing has been 
assumed respecting the constitution of the solid, and they all apply equally well 
whatever that may be. But in the reduction to the case of isotropism, it is im- 
portant to know the laws according to which strains are resolved in different 
directions ; or, to put more concisely a particular case of the general problem, 
to know what conditions must be satisfied that two systems of stresses, defined 
with reference to different sets of rectangular axes, may be equivalent. When 
the strains are of the first order of small quantities, the system (2s,,, 2s,, . . . Szy) 
follow the same laws of resolution as the stress system (S,....S,,). For in the 
case of isotropic media we have 
Sa = AO a2 2B « Sor y ey — Bs,, ? 
when 0 = s,, + S,, + s,, and is an invariant, and A, B are constants. 
But the same law holds good when the strains are not small, as I first found 
by actual transformation of co-ordinates. A simpler proof, however, is furnished 
by considering the strain ellipsoid, which as already shown may be written 
B+AtaA=eH+rypP+ 2+ 2 (S227 +... +5, 2y) = K’. 
Expressed with regard to new axes this becomes 
Be ty tea ar ty? 4+ 2742 (Spt? +... 4+ Sy LY) = K’, 
and the new strains (s’) are expressed /inearly in terms of the old (s). But 
d(uv w) 
d(x y 2) 
formation, it follows that the parts of the first and second degrees separately, 
and consequently the whole (s), follow the same law of resolution, which is that 
of strains of the first order already given. 

since the degree of any power of the distortions is unaltered by trans- 
