/'. i'i--fi/u>fit:/f Ft'furc of Equilibrium of Rotating Liquid. 147 



From this it results that the true azimuth 



of the Sun at the time of observation = N. 50 30' 54" E. 

 And since azimuth of Friar's Heel = 50 39 5 



2' of sunrise should be X. of Friar's Heel 811 

 Observed difference of azimuth 8 40 



Observed calculated - 29 



The observation thus agrees with calculation, if we suppose about 2' 

 of the Sun's limb to have been above the horizon when it was made, 

 and therefore substantially confirms the azimuth above given of the 

 Friar's Heel and generally the data adopted. 



" The pear-shaped Figure of Equilibrium of a llotating Mass of 

 Liquid." By G. H. DAUWIN, F.E.S., Plumian Professor and 

 Fellow of Trinity College, Cambridge. Received October 21, 



1901. 



(Abstract.) 



This is the sequel to a paper on " Ellipsoidal Harmonic Analysis," 

 presented to the Royal Society in June, 1901. 



Rigorous expressions for the harmonics of the third degree may lie 

 found by the methods of that paper, and the processes are carried out 

 here. The functions of the second kind are also found, and are 

 expressed in elliptic integrals. 



So much of the results of M. Poincare's celebrated memoir* on 

 rotating liquid as relates to the immediate object in view is re-in- 

 vestigated, with a notation adapted for the use of the harmonics 

 already determined. The general expressions for the coefficients of 

 stability having been found, those for the seven coefficients corre- 

 sponding to the harmonics of the third degree, as applicable to the 

 Jacobian ellipsoids, are reduced to elliptic integrals. 



The principal properties of these coefficients, as established by 

 M. Poincare, are enumerated. He has shown that the ellipsoid can 

 bifurcate only into figures defined by zonal harmonics with reference 

 to the longest axis of the Jacobian ellipsoid ; that it must do so for all 

 degrees ; and that the first bifurcation occurs with the third zonal 

 harmonic. 



A numerical result given in the paper seems to indicate that as the 

 ellipsoid lengthens, it becomes more stable as regards deformations of 

 the third degree and of higher orders, and less stable as regards the 

 lower orders of the same degree. 



* ' Acta Math.,' TO!. 7, 1885. 



