&66 CELESTIAL MECHANICS. I. vii. 29- 



q'^ + r^^) expresses the sum of the areas described in the 

 time t by the projections of the revolving radii of all the 

 molecules on the plane with regard to which this sum is a 

 maximum, and with respect to which N' and N" vanish. 

 For this plane we obtain, by making IV' and A" = 0, Oz= 



T» . J r / . / 1 sin ^ 



Br sm o — Jo cos (p, [or r smozzq cos (p, and =zta^ 



"* ■* cos<p 



=:■—], and Jq cos Q sin (p—Br cos Q cos <p + Q> sin firzO, 

 r 



[or — p' sin & z=: q' cos 5 sin ^ + r cos fi cos <p, whence 



sin 5 ^ ^ q . r . ^ ta^ 



= — ta 0:z: —sin ® H — -, cos <b : now sm ^ = z=l 



cosfi p p sec?) 



q' . r^ q , r 



i_ / —___—— 2 . and cos n: • con- 



seqnently — ta 5 zn --^ — I = -^ — ; — ^,1 whence 



1 p' p' . . 



cos 5= = — ,,-^ ,„ ;- ■=. -v, sm 5 sin ^ = 



seed VCp' + ^Hr^) k* 



— ^:^LJ- — i_. ^ ^ ; = — -i, and sm 6 cos ^ = 



By means of these equations we obtain the values of & 

 and p for any given time with regard to the plane of 

 greatest rotatory power. We have only further to deter- 

 mine the angle 4', made by x' with the common intersec- 

 tion of the fixed plane and that of the two principal axes 

 x" and y"y which requires a distinct integration. Now 

 since 5d^=d^J/. sin 6 sin (p — d5 . cos ^, and rdfrzd^' . sin d 



cos ^ . + dd . sin ^, we have qdit . sin fl . sin f + rdf . sin Q 

 cos ^ = d>|/ . sin H^ and d^^ z= — qd^t. -. — •?— ^ rdf. 



