480 PROFESSOR C. NIVEN ON THE 

The sum of the squares of the members of the several rows of V°, and the 
sum of the products of the corresponding members of each pair of rows, will be 
of great use to us in what follows. They are denoted by a, b,c,d,e, 7 These 
quantities also follow the same laws of resolution as stresses, a property arising 
from the fact that = 5 £ follow the ordinary parallelepiped law of resolution, 
just as wv w do, when the axes are rectangular. 
§ 10. Reduction to Isotropic Media.—The theory of the ellipsoid indicates 
directly that there are three invariant functions of the strains (s), of the first, 
second, and third degrees respectively. Calling these J,, J., J; it follows that 
all possible invariants of these three degrees are J,, mJj + nJ., mJi + n'IJ2 
+ pds. 
Also, 
Vig = [Big Spy Ses 
Jo = — 48y8., — 488 rp — 4S yaSyy + She + Sir + Shy l 
J = 48,8 yySez + SySra8ry — SoaSy2 — SyyStr — S28y 
(14). 
If we confine ourselves in W to terms of the second degree, we may take 
QW = mJ? + nd., 
4 
where m= k + F (See THomson and Tart, art. 682). 




The expressions for the stresses become very simple; for writing 
2W = (m — 2n)Ji + nJ >, where 
Jy SS eee 2S, 
V Seo = (M—QN)IT,. AW A MQW py UZ A Wy} Ho A Wy/UyUy) t 
: (15). 
A Sry = (M—2n) Ty. f + N(Wx. U0, + WyyllyVo-+ - - + Sry(UaP2 + UsUi)) 
in which a, b.../, have the values in the last paragraph of § 9, but now refer 
to V. 
These expressions admit also of the following transformations. 
V Sa = (m — 2n.J, —n)at+ ne +¢4+/”) 
: (15a), 
V Sy = (m—2nJ,—n). f+ nla + b.f+de) 
where also 
2,4+383=a+bd+ec. : (15d) ; 
or, with reference to the developments which follow in the next article, into 
these forms 
Vv Sex = (md, + Qn\a + ne =a 2 77a ) 
(150). 
V -Sxy = (mJ, + 2n). f+ n(de — cf) j 
