Page:Proceedings of the Royal Society of London Vol 69.djvu/155

Rh 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.

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.,' vol. 7, 1885.