STRAINED ELASTIC SOLIDS 495 



(d'^<l)/dr'^)o, (d^4)/drdr')o be the common values, assumed accord- 

 ing to [445] by the various first and second differential coeffi- 

 cients of (f) with respect to the variables ri, r^, rz. Use a similar 

 notation for x- Then if we write down 



Xo — 4>o, 



>ar/o \a 



r/o 



\ar2/o 

 \drdr'/o \ 



.drWo 



— V 

 drdr'/o 



we have four simultaneous equations to determine the four 

 quantities ^, e, f, h; these, as the text says, will give to the 

 approximations x, dx/dn, 5x/9^2, dx/dr^, . . . d^x/dridrz their 

 "proper," i.e., correct, values ^, d<i>/dn, d4)/dr2, dcjy/drs, . . . 

 d^(t)/dridr2 when n = r2 = n = ro, i.e., when the solid is in its 

 unstressed state not at the original temperature of the state of 

 reference but at the temperature for which it has expanded (or 

 contracted) from that state in the ratio ro. But by Taylor's 

 theorem, if we expand <f) in terms of ri — ro, r2 — ro, n — ro, 

 we have 



* = *, + ( ^ 



\dr 



■) (n - ,-.) + (^) (r. - r.) + ('-*) (ra - r,) 



i/o \3r2/o \9'Vo 



+ 1 {m (., _ r„)' + (q) in - r„y+ (^) (r. - r.)= 

 2! \\drVo \drl/o \drl/o 



+ 2 (-^) (r2 - ro) (rs - ro) + 2 (^^) (r, - ro) (n - ro) 

 \dr2dr3/o Xdndri/ q 



+ 2 ( — — ) (ri — ro) (r2 — ro) > + higher powers 

 \dridr2/o J 



= <^o + ( — 1 (ri + rg + rs - 3ro) 

 \ar/o 



