488 PROFESSOR C, NIVEN ON THE 
[Added 16th February 1876. | 
§ 16. The reduction of the work necessary to produce a compound strain 
in an isotropic solid has been effected, in the foregoing paper, on the hypothesis 
that the work necessary to produce a state of strain from perfect freedom, is a 
quadratic function of the component strains. But it may be useful to show 
how the same problem may be solved when the work is a function of any 
degree in these strains. , 
Calling xyz the relative co-ordinates of two particles of the unstrained 
solid, x,y,z, those of the same particles in the state of strain, it has been 
shown. that 
at+yita—(ev+y+e)=Aa’?+ By? +... +2Fay, . (25). 
where A=23 "3B =2s, 5...) see 
From this result the following invariants result— 
J=A+B+C 
H=D?+ E? + F’?—BC—CA—AB 
K=ABC +2DEF—AD’?—BE?—CF*”. 
And there can be no more invariant functions of the strains; for these 
equations are sufficient to determine the three principal strains in terms of 
J,H,K; and if there were a fourth invariant, it must be a function of these 
principal strains, and consequently of J, H, K. 
The work done in producing from freedom any state of strain must, there- 
fore, be a function of J, H, K of the form— 
W=4(mJ?+nH)+pJ?+qJH+7K+..., . (26). 
[The m here used is the same as that used by THomson and Tarr, p. 710, 
but the m is different. If the m there used be written m, we shall have 
m=4m—n]. 
We have now to find how J ,H, K for a compound system depend on its si 
component elements; and to do so, I use capital letters with suffix , to define 
the primary strains, letters without suffix to denote the secondary, and those . 
with suffix , to indicate the final state; while a, b,c, d, e, f denote, as formerly, | 
the primary quasi-strains. The corresponding invariants will also be similarly _ 
distinguished. We have, therefore— 4 
J=A+B+0C, J=A,.+Bo+Q, J, =A, +B,+C,,7=at+bte, 
and the remaining invariants H,H),....4 may be similarly written down. 
The following “mixed concomitants,” which make their appearance, are 
thus denoted :— 

