STRESSES DUE TO COMPOUND STRAINS. 489 
Jj =Aa+Bb+Cce+2Dd + 2Ke + 2E/ 
9, =A(bc—d’) + B(ca—e’) + ....+2F(de—cf) (27). 
— J2= (BC—D?’)(be—d’) +... .+2(DE—CF)(de—c/).* 
§ 17. We now proceed to find J,, H,, K,. If we turn to equations (9), 
we find they may be written 
1+A,=a;(1+A)+i(1+B)+... +20,8,F 
| F,=a,a,(1 + A)+6,6,(1+B)+...+(a8, + a8,)F . 
I. To find J, we have 
34+J,=a(1+A)+01+B)+...4+2/F, 
whence Ji=Sot+¥ 
panied (28), 
II. To find H,, it is most convenient to calculate, in the first place, (1 +.A,)? 
+(1+B,)?+(1+C,)?+2D{+2E}+2Fi, which is found, after a little reduc- 
tion, to be equal to 
Sa(1 + A)? +23(be+ @2)D? + 230?(1+B)(14+C) +43a/(1+ A)F 
+43¢/(1+ A)D +42 (ad+ef)EF , 
where = indicates summation extended to all terms of the same type. On 
expanding and substituting for A, + B,+C;, its value already found, we obtain 
H,=27+4—3+hI +29+ G+ 5: , 
and on putting A=B=...=0 we find also (29). 
H,=27+4—38 
III. To find K, we observe that K is the discriminant of the covariant 
(25), and that the problem of finding 1+ .A,,.... is the ordinary one of linear 
transformation, the modulus being V,; hence 
fee )(1+B,)(1+C,)+2D,E,F,\—...=V, ((1 +A)(1+B)(1+C)+2DEF-.. ”) 
Now, it is clear that, if Vj be expressed as a determinant, we shall have V; =4, 
* It is perhaps worth noting that there is still a fourth invariant, which depends on the systems 
(AB...F), (ab...f), namely, J =a(BC—D*)+0(CA—E?) + ...+2f(DE—CF), but it does not 
present itself in these investigations. It may be observed, however, that it is not really independent 
of the nine magnitudes J, H, K, j,,%, d,4,,4,. For every invariant relating to two systems of 
strains, or to a system of strains and one of quasi-strains, can depend on only nine elements,—the six 
principal strains, and the three magnitudes which determine one set of principal strain-axes with 
regard to the other. In fact, referring one set of strains to their principal axes, we see that the ten 
invariants involve only nine independent quantities. 
This is a special case of the more general theorem that n sets of magnitudes cogredient with 
strains give rise to n(2n +1) invariants apparently independent, but of which only 6x—3 are actually 
independent, the remaining 2n?— 5n+ 3 being functions of these. 
VOL. XXVII. PART IV. 6 0 
