124 Sir W. Thomson on Cauchy's and Green's 



E = <r(A7X + Bmfi + Cnv) + \a- 2 { L + M + N + (A - L)\ 2 

 + (B-M)ya 2 + (C-N)v 2 -LP-Mm 2 -Nn 2 + 2[^2G + L-M-N)/ 2 ; 

 +-(2H + M-N-L)mV + (2I + N-L-M)nV][ 



16. To interpret this result statically, imagine the solid 

 to be given in the state of homogeneous strain (a, /3, 7) 

 throughout, and let a finite plane plate of it, of thickness h, 

 and of very large area Q, be displaced by a shearing motion 

 according to the specification (3), (4), (5), (6) of § 8 ; the 

 bounding-planes of the plate being unmoved, and all the solid 

 exterior to the plate being therefore undisturbed except by the 

 slight distortion round the edge of the plate produced by the 

 displacement of its substance. The analytical expression of 

 this is 



" =/(/>) (26), 



where/ denotes any function of OP such that 



*' j J 



dpf(j>)=0 (27). 



If we denote by W the work required to produce the supposed 

 displacement, we have 



w 



■■Q^dpSR-VW .... (28), 



SE being given by (25), with everything constant except <r, 

 a function of OP; and <W denoting the work done on the solid 

 outside the boundary of the plate. In this expression the first 

 line of (25) disappears in virtue of (27); and we have 



W — W "1 



n =i{L + M + N+(A-L)X 2 +(B-M> 2 + (C-N)v 2 

 ^ I 



-LP-Mm 2 -"Nn 2 + 2[(2G + L-M-N)/ 2 \ 2 > (29). 



C h 

 + (2H + M-N-L)mV+(2I + N-L-M)nV]}l dp* 2 j 



Jo J 



When every diameter of the plate is infinitely great in com- 

 parison with its thickness, q/JjQ, is infinitely small ; and the 

 second member of (29) expresses the work per unit of area of 

 the plate, required to produce the supposed shearing motion. 

 17. Solve now the problem of finding, subject to (5) and 

 (6) of § 8, the values of /, m, n which make the factor { \ of 

 the second member of (29) a maximum or minimum. Tliis is 

 only the problem of finding the two principal diameters of 

 the ellipse in which the ellipsoid 



