704 Prof. P. Lowell on the Limits of the 



9. Substituting in equation (7) for a its value in terms of 

 h and <?, reducing and eliminating b from both sides of the 



equation, we have, calling "-^ =§ , 5 



P 



S(l-e 2 ) (3-2e 2 ) v'l^ 2 



sin- 1 * + 2=0. . (8) 



Since q is known from the rotation period and the mean 

 density, we can deduce the oblateness tj consonant with 

 homogeneity in series as follows :— 



. . e z 3/ , Ue 7 35^ 9 



gm -i, =e+ _+_-+_ +n ^ + , 



also since <r= s— 



a 3 



77 = ^ and e, the ellipticity , = — =— - , 



we have e 2 =2 v - v 2 = 2€-3e 2 + 4,6*-, 

 whence l — e 2 = l — 2r)-\- v 2 = l — 2e + 3e 2 -Ae^ + 

 and Vl-£ 2 = 1-?7 = 1 — e + e 2 -e 3 + . 



Making the various substitutions in the equation (8) we 

 get 



8 4 2 250 3 

 ^ = l5 ?? -35 77 -1641* appr ° X - ' • ■ I 9 ) 



8 68 , 61 „ 



= ro € ""i05 6 + roo e a PP rox - 



approx. referring here to the third term. 



10. We may now reverse the series by the method of 

 undetermined coefficients and get rj or e in terms of q. 



If q = a x r\ + a 2 ^ 2 + a 3 if } &c, 



then if the series be convergent 



r } = b 1 q + b 2 q 2 + h 3 q i &c. 



and we get 



(aA— l)q+ [a\h + « 2 V) ? 2 + (%& 3 + 2«A&2 -f <V>i 3 )<? 3 > &c. = 0. 



Since # may have any value within the limits of con- 

 vergence, the coefficients of this series must be identically 



