STRAINED ELASTIC SOLIDS 409 



from a right angle. The shears of lines parallel originally to the 

 axes OY' and OZ', and of those parallel to the axes OZ' and OX', 

 are likewise given to a close approximation by 64 and e&, re- 

 spectively, or practically 623 + 632 and esi + 613. 



Now we know that there is one set of axes of reference, for 

 which there is no shear. Suppose we had chosen them at the 

 outset and carried through the analysis just finished, then three 

 of six strain functions calculated as in (9) would be zero, viz. 

 the three indicated by the suffixes 4, 5, 6, To make this as 

 definite as possible let us indicate these three principal axes of 

 strain by OL', OM', ON', and let the coordinates of Q', relative 

 to three local axes through P' parallel to these, be denoted by 

 the letters X', ij.', v'. We should arrive at a result similar to (8) 

 viz., 



,2 



P"Q'> = e,x'2 + e2^'2 +63 /2 + 2un'v' + 265/X' +2e,\'n', 



where ei, €2, cs, etc., would be six strain functions such that 

 (ei)i would be the ratio of elongation parallel to OL', etc., and 

 also such that the cosine of the angle between two lines originally 

 parallel to OL' and OM' would be ee/Ceieo)*. But as this angle 

 still remains a right angle, ee would have to be zero and simi- 

 larly for €4 and €5. Hence we would arrive at the result 



'2 



P"Q"" = e{K" + 62^'^ + e^v 



In his discussion Gibbs indicates the three "principal ratios of 

 elongation" by the letters n, r2, n, so that his notation and ours 

 are connected by 



ei = ri^, €2 = ra^, ea = n'^. 



Certain relations, very necessary to our progress, between the 

 €r and the e^ symbols can now be obtained very elegantly by an 

 artifice depending on a theorem concerning quadric surfaces 

 quoted in the Mathematical Note. Keeping P' as our local 

 origin, allow Q' to move about on a locus of such a nature that 

 the corresponding positions of Q" lie on a sphere of radius h 

 around P" as centre. By (8) we see that the equation of this 

 locus in the ^', 77', f ' coordinates is 



eir^ + eov" + e3f'- + 2eW^' + 2e,^'^' + 2e,^'rj' = h\ 



