STRAINED ELASTIC SOLIDS 493 



The justification of [445] can be easily given as follows. Re- 

 membering that i^F' is a function of E, F, H, say ^{E, F, H), it 

 follows that 



dypv _d4> BE d4> dF d<i> dH 

 dri ~ dE dn dF dri dH dn 



dE dF dH 



Similarly 



^ = 2r. % + 2r. (rl + r?) ^ + r^n -^• 

 ara dE dF dH 



Obviously 



dxf'v' _ d\f/v' 

 dri dri 



if ri = Ti = rs, and exactly similar arguments cover the other 

 equations. The wording of the argument at this point on page 

 212 is a little confusing; for, as the text itself points out, this 

 theorem is true "if i/^' is any function of t, E, F, Hj" not merely 

 the approximative linear function of [444] ; then just lower down 

 we have references to "proper" and "true" values of ^pv. It 

 might be better therefore to introduce two functional symbols 

 one <i){t, E, F, H) to refer to the "true" value of ypv and one 

 x{t, E, F, H) to refer to the linear function of E, F, H in [444] 

 which is approximately equal to ypv. These can both be 

 expanded as series in terms of ri, r2, r^, or rather of ri — ro, 

 ^2 — ro,rz — ro; the discussion centers round the problem of deter- 

 mining at what power of n — ro, etc., the two series begin to 

 show a difference. A little thought will show that the series for 

 X will terminate at fourth order terms. In fact writing for 

 the moment x for ri — ro, y for ra — ro, z for rs — ro, we see that 



X = i + e{{x + roY + (2/ + ro)^ + (z + ro)'} 



+ f{{y + roYiz + roY -\- (z-h roYix + ro)^ + 



(x + roYiy + roY] 

 + h{x + ro) {y + ro) {z + ro). 



