—— 
STRESSES DUE TO COMPOUND STRAINS. 481 
There is a remarkable corollary from these equations. There exists, as we 
shall see in § 13, or as we may deduce from the laws of transformation of 
a...f, one set of rectangular axes at each point for whichd=e=/f=0. If 
the solid be homogeneous after the strain, these have the same direction at every 
point. It follows from the above equations that for these axes there is no tan- 
gential stress, and the normal stresses are the sums of two parts of which one 
is directly as the corresponding a, 0, c, and the other part as its square. And, 
generally, at every point of an isotropic solid the stresses (each multiplied by V7 ) 
are functions of a.../, which we may call the quasi-strains. 
§ 11. The calculation of w’ will usually be a very long and troublesome 
matter ; for it contains 21 terms of the form D),°Dy,, each of which terms Dj, 
contains 21 terms of the form oj. We should thus have, in general, to take 
account of 441 terms. But if we narrow the problem, as in last article, by taking 
W to consist of terms of the second degree only in (s), and by supposing it 
isotropic with regard to these terms, w’ appears reduced to a form of remarkable 
and rather unexpected symmetry. 
When W = 4(s) is homogeneous and of the second degree in (s), then also 
V 00 == 'o( DF 
= }(mIj + al,) gibi 
where I, and I, are the invariants of (D) 
If for 20,,, 20,,... 0, we write A, B...F, we find 
ee eee arte yey deny 12) He 
The calculation of I, is most easily performed by find finding I, + 2Ij; 
when this work, which naturally is not devoid of symmetry, is gone through, 
we arrive finally at 
E (BC—D®) (be—@) + (CA—E®) (ca—e?) + (AB—F*) (ab —f?) 
aa i +2(EF — AD) (¢f—ad) + 2(FD —BE) (fd —be) + 2(DE—CF) (de—ef) I 
In these results a... fare the initial guasi-strains. 
-{§ 12. The strain ellipsoid corresponding to the second strain may be 
written 
Aa’ + By? + Cz? + 2Dyz + QHhzx + 2Fary = 1 ; (6), 
while the quasi-strain ellipsoid corresponding to the initial strain is 
au” + by? + cz? + Qdyz + Qezx + 2fay = 1 (0). 
Let the discriminants of these surfaces be S = 2J} ands, and let them be 
VOL. XXVII. PART Iv. 6 M 
