406 RICE ART. K 



the so called "principal axes of strain," which are not only 

 mutually orthogonal before the strain, but remain so after it, 

 although in general they are not pointing in the same directions 

 after as before. This is a result used by Gibbs and demon- 

 strated by him in a different manner (Gibbs, I, 205 et seq). On 

 page 204 also occurs the sentence: "We have already had 

 occasion to remark that the state of strain of an element con- 

 sidered without reference to directions in space is capable of 

 only six independent variations." This remark is illustrated 

 by the result which we have just obtained, since although there 

 are nine strain-coefficients, the strain, apart from the rotation 

 which produces no relative displacement of neighboring parts, 

 depends on the six quantities 



€l\, 622, 633, 



^23 + ^32 631 + ei3 ei2 + 621. 

 , , 



Gibbs then continues: "Hence it must be possible to express 

 the state of strain of an element by six functions of dx/dx', . . . 

 dz/dz', which are independent of the position of the element." 

 The functions chosen by Gibbs are not so formally simple as 

 those written above and have a certain appearance of arbitra- 

 riness about them. So we will address ourselves to the task of 

 explaining how the six functions defined in [418] and [419] 

 naturally arise in a further discussion of strain. Indeed, 

 the whole of the material treated in Gibbs, I, 205-211 may 

 prove troublesome to follow without some help over analytical 

 difficulties, which will now be given. The treatment which 

 follows will present the matter from a somewhat different angle 

 and at the same time bring out the physical nature of the er» 

 coefficients. 



Let us revert to equations (3) and use them to determine the 

 length of P"Q" as a function of the local coordinates of Q', the 

 original position of Q", with reference to the axes through P', 

 the original position of P". It is easy to see that 



p"Q" = r" + v'" + r' 



= e,^" + e^v" + esf'^ + 2e,v't + 2e,^'^' + 266^^?', (8) 



