464 



RICE 



ART. K 



of Xy to Yx, Yz to Zy, Zx to Xz is one of those succinct, sweep- 

 ing statements which he makes from time to time with complete 

 justification, but with a whole array of intermediate steps in the 

 reasoning omitted, to the bewilderment of the reader not so well 

 versed in analytical processes. It was in \ iew of the awkward 

 situation at this point that we have in our discussion introduced 

 and defined Xx, Xy, . . . Zz first, treating them in a manner 

 which will have been familiar to any reader acquainted with 

 modern texts on elasticity, and have already proved the 

 equality of Xy to Yx, etc. Later, it will be recalled, we intro- 

 duced Gibbs' more general stress-constituents Xx', Xy', . . . Zz' 

 and gave some care to their precise definition and to the equa- 

 tions (38) which connect them with Xx, Xy, . . . Zz. It will 

 be apparent from these equations that in general Zy is not 

 equal to Yx', for example. Let us, however, make the two 

 sets of axes coincide so that an becomes en, etc., and ^^s, the first 

 minor of Urs in the determinant | a \ becomes Ers, the first minor of 

 Crs in the determinant \e\. Equations (38) will be replaced by 

 equations in which Ers is substituted for A rs. Even so, as we 

 pointed out earlier, Xx' does not become identical with Xx, etc., 

 unless the difference between the state of reference and the state 

 of strain is so little that a rectangular parallelopiped in the one 

 is but little distorted from that shape in the other. To elabo- 

 rate this latter point a little more, it will be observed that in 

 such a case the determinant 



en ei2 eis 

 621 622 623 

 631 632 633 

 approximates to the form 



1 



612 

 1 



— 612 



— ei3 — ^23 



for en, 622, 633 are little different from unity, and 623 + 632, etc., 



ei3 



623 

 1 



