STRAINED ELASTIC SOLIDS 413 



to the plane of P'QiQi, and a", /3", 7" those of the normal to the 

 plane of P"Qi"Q2", we have the following results: 



K'a' = KVf/ - 172'f/), K"a" = Km"r2" - ^2"h"l 

 K'p' = Kf:'^2' - f2'^/), K"^" = Kfi"e/' - r2"^/'),[ (17) 

 K't' = Ka'ri2' - ^2'm'), i^"7" = m"V2" - ^2"vn.j 



If one now uses equations (16), and is careful to keep to the 

 convention about the signs of the first minors as explained in 

 the note, it is not very troublesome to prove that 



m"^2" - W'^x" = Enim'h' - ^2'fi') + ^i2(fi'^2' - r2'^/) 



+ £'13(^/^2' - ^2'm'), 



and two similar results which can be succinctly written 



K"cx" = K'(Ena' + E,ol3' + Enl'V, 



K"fi" = K'iEW + ^22/3' + EnV),\ (18) 



K"y" = K'(Ez,a' + ^32/3' + ^33t').. 



These are essentially the steps by which one passes from 

 equation [381] to equation [382], K' and K" being the Ds' and Ds" 

 of Gibbs. (There is of course at the moment some restriction 

 on our Brs and E^ symbols, i.e., our differential coefficients and 

 the determinants constructed from them, due to our restriction 

 as to the axes chosen in the strained system; we have already 

 referred to this and it will be removed shortly; for the moment it 

 involves us in the use of doubly accented symbols such as ^", 

 K", a", etc., so as to avoid confusion later when we widen our 

 choice of axes.) 



The interpretation of the quantities En as determining super- 

 ficial enlargement caused by the strain is very clearly indicated 

 in (18), and a very elegant analogy can be exhibited between 

 equations (18) and the equations (3) in which the ers quantities 

 obviously determine finear enlargement. To this end we 

 remind ourselves that an oriented plane area is a vector quantity, 

 and is therefore representable by a point such that the radius 

 vector to it is proportional to the area and is parallel to the 

 normal. Thus the triangle P'QiQ/ can be represented in 

 orientation and magnitude by a point whose coordinates are 



