STRESSES DUE TO COMPOUND STRAINS. 475 
in each pair of columns. They are denoted by §,., Sy) Sy Syz, Sx. Somewhat 
further on, in dealing with long formule, I shall write them simply 4A, 4B, 4C, 
D,E,F. They are readily expressible in terms of the edges of the tetrahedron 
PK’. 
The infinitesimal strained sphere x? +y/? +27 = K’ was produced by straining 
the ellipsoid 
S 
2X29 
B+ Yi t A+ WAS + SyYit ... + Sy. %y,)=K.  . . (3). 
[(@, y, 2,) are the relative co-ordinates of a point very near to P]. 
This result will be afterwards useful in determining the laws of resolution 
of a strain-system from one set of axes to another. 
Equation of Energy. 
§ 3. The energy required to strain any element dx dy dz into its final form 
is W dx dy dz, where W is a function of the strains which, as will be found, 
cannot be of a lower degree than the second. 
Let S,2; Sy, -- + Sz, be the six stresses at any point considered as tractions. 
The work on any triangular element, and so on any element d>’ of the strained 
surface, due to an infinitesimal displacement dw, dv, dw is the sum of the quanti- 
ties of work done in its projections on the three co-ordinate planes. Its value 
is therefore 
Ai(S,,0u +S,,80 + 8,,0w) + Aj(S,.0u+5,,50 + 8,,6w) + AXS,,6u +8,,60 + 8,6). 
The work done over the surface of a strained body is therefore 
dy dz\(Sox-U,+8.yV,+8,,.W,) du + two other similar terms} 
+ //dz dx {three similar terms} + //dx dy {three similar terms} . (4). 
Now the work done by internal stresses is of course ///OW . dx dy dz. 
adW 
Say 
And awa ss. + OS) ten age 
and OS cn = UjOU, + V,00, + WOW, [ ie (5). 
OS) = UgOUy + Vg0U. + WOW, + UUs + V.OUs + WOW; J 
The work done by the impressed forces=p, ///(X6u + You + Zéw)dax dy dz. 
That consumed in producing motion=p hhh Gg ae EUS jet 72 eS y ae Tsu \de dy dz. 
On substituting these values in the equation which expresses the conserva- 
tion of energy, and equating according to LAGRANGE’s principles the coefficients 
of 8u... to zero, both throughout the interior and over the surface, we obtain 
