Page:Proceedings of the Royal Society of London Vol 4.djvu/58

Rh and the corresponding maximum value of 03370, and re- marks that, with respect to spheroids of revolution, it thus ap- pears that an equilibrium is impossible when g, or its value in terms of the density, is greater than 0-3370. In the extreme case, when is equal to 0-3370, there is only one form of equi- librium, the axes of the spheroid being g k and k 1 (25293)2 or 27197 k; but when is less than 0.3370 there are two different forms of equi- librium, the equatorial radius of the one being less, and of the other greater than 2-7197 k, k being the semi-axis of rotation. The number of the forms of equilibrium in spheroids of revolution, he remarks, is purely a mathematical deduction from the expression of the ratio of the centrifugal to the attractive forces; and as this has been known since the time of Maclaurin, the discussion of it was all that was wanted for perfecting this part of the theory. Returning to the general equations of the problem, the author deduces the equations

g= d p d o is a definite integral, such that where do' d2 .A, g d2 p A and (A-)*,

which equations apply exclusively to ellipsoids with three unequal axes, and solve the problem with regard to that class. From these he derives another equation, which he states is no other than a trans- formation of his first fundamental equation, and is equivalent to other transformations of the same equation found by M. Jacobi and M. Liouville

He also remarks that a limitation of one of the constants, which the verification of this formula requires, agrees with the limitation of M. Jacobi; and further, that the relations which may subsist be- tween the constants proves that there does exist an infinite number of ellipsoids not of revolution, which are susceptible of an equili- brium.

After determining the corresponding limits of these relations of the constants, p being contained between the limits 1-9414 and 1, while increases from zero to infinity, he remarks that an elliptical spheroid formed of a homogeneous fluid can be in equilibrium by the action of a centrifugal force only when it revolves about the least axis.

He next deduces the general value of g (the ratio of the forces), and thence its value in one extreme case, when r 0, or when