478 PROFESSOR C. NIVEN ON THE 
The energy due a compound strain may be readily found, for let W° corre- 
spond to the initial strain, and W to the final, 
W® = ¢(s°"), W = G(s? + D). 
Whence by Tay tor’s theorem, 
dW? iL d2W° d2W? 
ae so eee 5) az Die + 2agga, DeDay + «- \ E 

W-—W°=D 
If V° correspond to the first strain, dvdydz V° = d&dydl; substituting 
which, and remembering the expressions given in equations (7) for Sy,, it 
follows that 
20 
Wade dy de=Wdx dy de+ (S%jou-+... +820, dE dn dt + su ae Deaton) . dédndt (10). 
The two last parts of this expression I shall usually write W’d&dydf and 
w'dé dn df respectively. 
[§ 7. Without assuming a knowledge of (7) , the second member of (10) may 
be written 
We Sey FS. e Te, 
where the forms of Ti, are known. If we now suppose the second strains to 
be very small, and after differentiation, according to the formula (8), which 
are well known, put w = v = w = 0, we obtain 
Tye = Bras 
which furnishes a new proof of the forms for S,, . | 
§ 8. The system of additional stresses (S — S°) introduced by the second 
strain, which we shall always suppose to be small and of the first order, can 
now be found. The part of it which arises from w’ may be found from equa- 
tion (8). 
We have in general, on putting V=V°. V’, 
wy 

di 
V’ Sau = S7(Gi,2 + Say + Oye) Oust One t On) 8 + gon | 
*\-\ amt 
ape du | dv | dw | 
ee GE | da tb ge J 
In particular, 
du dv | dw du du du dw 
NO 2 0 wes Pa aes ial i 
S..~— St, =—S2. (Ge + 7, + ae) +2(GeSt + F, Sh + Ge Sh) Fac | 
(12). 
dw dw’ | ) 

dv du dv du 
a (iY vee Or a eee 0 : 0 
Say — Sty = Sty. gy + gg: Ste + GS + ae Su + ae Sy + 
deny ) 


