Forces derivable from a Potential of the Second Degree. 53 



yy and yz may be deduced from xx and zx respectively by inter- 

 changing P with Q, x with y, and a with 6. 



If P, Q, and R be of the same order of magnitude, the principal 



terms depending on them in the values of xx, yy, and yy are likewise 

 of the same order ; but in zz, yz, and zx, the principal terms in P and 

 Q are only of the same order as the secondary terms in R. I have 

 thus thought it best, in (47) and (49), to retain secondary terms in 

 the coefficients of R, and to write the second approximation value of 

 n. If, however, R be zero, the terms on the left-hand sides of these 

 equations with <? in the denominator should be dropped. 



When R alone exists, or the bodily forces are parallel to the long 



dimension, then, except near the ends of the long axis, zz is large 

 compared to yz and zx, while these in their turn are, at most points, 



large compared to xx, yy, and xy. The hypothesis usually made in 

 treating long rods, viz. : 



xx = yy = xy = (50), 



is thus approximately true, dxx/dx is, however, of the same order 



as dzxjdz, and dzxjdx of the same order as dzzjdz, so that the neglei t 

 of any of the nine differential coefficients appearing in the body-stress 

 equations would be unjustifiable.* 



When the bodily forces are perpendicular to the long dimension, 



then excluding special values of x, y, z xx, yy, zz and xy are of the 



same order of magnitude, and are large compared to yz and zx. This 

 result differs widely from (50). 



When P, Q, R are of the same order of magnitude, we may, for a 



rough first approximation, neglect all the stresses but zz, and take 



= fly = - 



(51). 



fa/c = R/>c 2 /E J 



The strain at any point is the same as in a long bar subjected to a 

 tension at its ends equal per unit of section to the local value of zz. 

 * See Todhunter and Pearson's ' History/ TO!. 2, Part II, pp. 189191. 



