STRESSES DUE TO COMPOUND STRAINS. 485 
Def. i. Let 2, y, 2 be rectangular co-ordinates, and (w, v, w) a ternary 
group such that for rectangular axes, 
un + vy + we = an invariant, J not =0 . : (23), 
then (wz v w) constitute a force group. 
Def. 2. Let (a, 6, c, d, @, f) be a sextic group such that 
ax” + by? + cz? + Qdyz + 2ezr + 2fay = an invariant, J’ (24), 
where J’ differs from zero, then (a,...,,/) from a stress group. 
The known properties of planes and quadries permit us at once to: 
enunciate. 
Theorem (a). For an infinite number of sets of rectangular axes, one axis of 
which is fixed, the ternary group (uv, v, w) becomes (w’, 0, 0), and 
the axis of 2’ is the axis of the group; so for one set of rectangular axes 
the sextic group becomes (a’, 0’, c’, 0, 0, 0), and the axes of a’, 9’, 2’ 
may be called the axes of the group. 
Since in the above definitions the quantities w..a...are involved linearly, 
we derive the following 
Theorem (b). From two force- or two stress-groups we may derive new force- 
or stress-groups by taking the sums or differences of the corresponding 
constituents of the groups for constituents of the new group. 
Since v.4+y.y + %.2=7", an invariant, it follows that the characteristic 
law of resolution of (7, y, 2) is itself that of forces; and since w#. a + 7.9” 
+ 27.27 + Qyz.yz + 2ex. cu + Quy. ay =7* an invariant, it follows the group 
(2’,y’... ay) is itself a stress-group, and has the corresponding characteristic 
law of resolution. Hence follows, 
Theorem (c). If (wu, v, w), (U, V, W) be two force-groups, then wU 
+0.V +w.W is an invariant J; and conversely if wU + oV + wW 
be an invariant, and one of the groups be a force-group, so also is the 
other: Also if (a....f) (A....F) be stress-groups, then is Aa + Bb 
+ Cc + 2Dd + 2Ee + 2Ff/an invariant; and conversely, if this expres- 
sion be an invariant, and one of these groups be a stress-group, so must 
the other. 
The theory of planes and quadries allows at once to write down the funda- 
mental invariants of each group, as we may do in 
Theorem (d). The invariant of (wu v w) is I = uw? + v* + w’; and the invari- 
ants of the stress-group (A.... F) are 
J,=—A+B+C,J,=BC+CA+AB—D?—E?—F?, J,= ABC +2 DEF—AD°?—BE?—CF?. 
The following are a few of the applications of this theory in various depart- 
VOL, XXVII. PART IV. 6N 
