STRAINED ELASTIC SOLIDS 411 



being essentially the third equation of [439]. Our equations dif- 

 fer from those of Gibbs in the greater generality which he adopts 

 concerning axes of reference before and after strain. But this 

 restriction we shall be able to eliminate presently, with no great 

 trouble. In the meantime let us continue with the other two 

 equations in (11). A glance at (9) shows that the first is just 



en^ + ei2^ + ei3^ + 621^ + 622^ + ^23^ + eai^ + 632^ + 633^ 



= ri2 + Ti" + n\ (14) 



The second of (11) gives a little more trouble; but the reader 

 may take it on faith, if he does not care to go through the 

 straightforward algebraic operations, that the following result 

 can be verified. If one squares the nine first minors of the 

 determinant (12) and adds them then the sum is equal to 



€263 + 6361 + 61^2 — €4"^ — 65^ — e^,^. 



(A less tedious method of showing this would have involved us 

 rather too deeply in the theory of determinants.) Hence, by 

 the second equation of (11), 



En' + £"22' + i?33' + £"21' + £"22' + Eiz' + En' + ^32^ 



+ £33' = raVs^ + nW + nW, (15) 



where we are representing the first minor of en in the determi- 

 nant of the ers by En, that of 612 by £'12, and so on. (The use of 

 this double suffix notation is obviously of great convenience at 

 the moment. The Ers used here must not be confused by the 

 reader with the symbol E used by Gibbs without any suffix, to 

 which we will be referring presently.) Equations (14) and (15) 

 are essentially the first two of the equations [439]. 



If we consider a rectangular parallelopiped whose sides are 

 parallel to the principal axes and each of unit length, we know 

 that it remains a parallelopiped after the strain (although it 

 may be rotated) and its sides become n, r^, rs, respectively. 

 Hence nriTs is the ratio of enlargement of volume, and so we 

 see that this is a physical interpretation of the determinant (12), 

 while the determinant in (11) is of course equal to the square of 

 that ratio. Further, the sum of the squares of the nine first 



