328 Prof. G. H. Darwin. On Jacob?* Figure of [Nov, 25, 



If a. be infinitely small, so that the Jacobian ellipsoid of three 

 unequal axes becomes in the limit an ellipsoid of revolution, we have 

 7 given by 



_ sin 27— T 5 g- sin 47 

 ^ 1— \ cos 47 



The solution of this is 7 =54° 21' 27". 



If we write tan 7 = /, as in Thomson and Tait's ' Natural Phi- 

 losophy,' §778', this equation becomes 



tan~V _ 1 + -V-/ 2 



which is the equation (9), §778', of that work. 



The ellipsoid of revolution of which the eccentricity is sin 54° 21' 27" 

 belongs to the revolutional series of figures of equilibrium, and is 

 the starting point of the Jacobian series of figures. As shown by 

 Sir William Thomson, it is the flattest revolutional figure which is 

 dynamically stable. The Jacobian figures of equilibrium are initially 

 stable, and as stated by M. Poincare,* there is for this value of 7 a 

 crossing point of the two series, and an exchange of stabilities. 



If a. be small, it appears that sin a. is given by 



sin** = 10'9266528 sin ( 7 _54° 21' 27"), • • • (21) 



and u by 



~ = £ cot 3 7 [ (3 + tan 2 7 ) 7 - 3 tan 7] 



+ h^lti*™^ • (22) 



Approximate Solutions of the Problem — continued. 



Now A = p^l + ^ 2 Q 2 ... 



Therefore 



A [ (1 + Q2) j^_j4V] =^j(| + Q*)-§(l + QS) 



V|/[ A (| + ^tQ')-h>-H}- 



Hence 



which is the equation (24) in the text. 

 * " Sur 1'lSquilibre d'une Masse de Fluide, &c," ' Acta Mathematica/ 7, 1886. 



