442 RICE 



ART. K 



constituents Xx, etc., are not to be confused with the stress- 

 constituents Xx etc., of customary elastic sohd theory. Gibbs 

 himself gives on page 186 a physical signification to his constit- 

 uents, which brings home to the careful reader how essential 

 it is to be on guard when it is a question of giving a measure of a 

 physical quantity -per unit of length or area or volume. His own 

 statement is so brief that for clarity it can be somewhat ex- 

 panded. He asks us to consider an element of mass which 

 in the reference state is rectangular (a "right parallelopiped" as 

 he calls it) with its edges parallel to the axes OX', OY', OZ'. 

 We shall adopt a method similar to that employed previously 

 and regard the center of this at a point P', whose coordinates 

 are x' , y', z'. The middle points of the faces perpendicular to 

 OX' shall be named Q' and U', the coordinates of Q' being 

 x' + ^', y', z', and of U', x' - ^', y' , z'; and so on. (The dx', 

 dy', dz' of Gibbs are 2^', 2r]', 2^'.) In the strained state the 

 element is in general an oblique parallelopiped the center of 

 which is at P, whose coordinates are x, y, z with reference to 

 the new axes OX, OY, OZ. The coordinates of Q, the displaced 

 position of Q' , and still the center of one of the faces (now a 

 parallelogram), are a: + ^, ?/ + 77, 2 + f , where 



k = ank', 



r = asir. 



(See equations (21), noting that the local coordinates of Q' in 

 the local axes at P' are ^', 0, 0.) Now consider a further infini- 

 tesimal displacement from this state in which only an varies, but 

 not any of the other eight strain-coefficients. In such a varia- 

 tion ^ will alter by ^ • 8an but 77 and f will not vary; i.e., the face we 

 are considering will move further from the center of the element 

 in the direction of OX (as Gibbs postulates in line 12 of page 

 186) by an amount ^' • 8an. Similarly the face opposite will move 

 relatively to the element's center an equal distance in the 

 opposite direction; in other words one face will have separated 

 from the other face by an amount 2^'oaii. Hence the work done 

 by the components of the force on the element across these faces 

 parallel to OX is equal to the product of 2^'aaii and this force. 



