482 PROFESSOR C. NIVEN ON THE 
reciprocated with regard to the spheres whose radii are TE and. F ; thence we 
obtain 
es SS ae ean iggy 
(be —d*)x? +...+2(de — f)ty=1 
2I, and — I, are the invariants of the first order of the first and second pairs 
of surfaces |. ; 
§ 13. If the body after the first strain be homogeneous, and we choose the 
co-ordinate axes parallel to these of the quasi-strain ellipsoid, we may put 
d=e=f=0, anda, b, ¢ will be constant throughout. 
Thus, 
Lo = dex + boy, +.¢o, (19. 
I, = be(o,2 — 40,02.) + 66.2 — 400622) + AO — 40 r2F yy) 
We are now in a position to determine the plane waves which can be pro- 
pagated unchanged through such a solid. 
[Inserted 16th February 1876, and following the notation of § 16, 18. 
_The equations of motion are as follows :— 
; d? ad? d? 
Putting Vi=aznt ony +675 
d? a? d? 
aa © Yeas Poe Agee 
Vi= Pat Vat hae 
we find, after some reduction, that the equations take the following forms: - 
dy d¥ 
P ap =U(V.+4Vi)utemz 
a? d4¥ 
Pp sas +bVi)o+ amg, . ° (20.) 
a d 
pow=n(V,+eViw+dmq 
where m=4m—n. 
To solve these, put 
U,0,W=(U,, %, W,) SIM = (a+ gy + cz—Vi) , 
where p+ 7t+r=l1, 
and let D,=ap’ + bq +¢r? 
DoS Pa ae Q’q? + Rr’ 
p NaS pe 
Nn 2 
0=apu, + bv, +¢/'w, 

