468 S. W. Gibbs—Double Refraction and Cireular 
that is, of the ratios of the quantities a, f,,7,, 4, Be ee ior 
not affect the period by any quantity of the same order 
magnitude.* ‘These variations must however be consistent oe 
equations (4), which require that 
Lda,+Mdf,+Ndy,=0, Lda,+Md6,+Ndy,=0. (17) 
Hence, to obtain the conditions which characterize free vibra- 
tion, we may differentiate mi jemi (16) with respect to on Bis 
Ty 4 Ax 7» regarding all other letters as constant, and give to 
da,, ap,, dy,, da,, dP,, are sabe values as are eofitet ak 
equations (17). “Now dj,, dy,, are independent of da,, df,, 
ay and ei oo ihre variations, values Meygebcy es either to 
» Ty OF to @, B,, 74) are pos ssible. If, then, we se rentiate 
‘i aasii it) ae respect to a,, 8,, 7,, and substitute first a 
» Zy and th st 4,, Bo 7 for da, dP,, dy, and also difforentiata 
with respect to a,, f,, 7,, with similar substitutions, we shall 
ster all the Uilepetident equations which this principle will 
"dese 
If we differentiate kp respect to a, f,, 71, and write a, A, 
7: for da,, df,, dy,, we obtain 
aa, +bf,'+ey,+ep,y,+fy.a,+9%P, 
+P P(L(B,y,—y,8.) +M(y,a,—4a,y,)+N(a,8,—f,a,) | 
Pr 
Sales Pets): (18) 
If we differentiate with Bhi to a, Bi, 71, and write a, Py, 72 
for da,, dB,, dy,, we o 
2aa,a, + 28,8, + 2cy,y.+ (BY ,+ VP.) +P(Vi 8+ 42) 
2 
+9(a,h,+ fa.) ~o (a,a,+ BB, + VV 2) (1 9) 
If we differentiate with Mages tO Gd, Po, 72, and write dy, 
for da, dB., dy, we obta ze nats 
aa? +bB)+cy,'+ef,y,+fya,+94,f, 
+PPIL(B,y,— 7.6.) +M(y,4,—a,y,) +N(a,8,—f,a,)] 
=S(a +BS +7, )- (20) 
The equation derived by dilconishar with respec Qn, Po 
Y2, and writing a, f,, 7; for da,, dP, dy2, is identical “eth “t19), 
We should also phactes that binetious (18) and (20) by addi- 
* Compare § 11 of the former paper, page 270 of"this’volume. 
