326 Royal Society. 



given. On these suppositions all the level surfaces are determined, 

 and the problem is solved, by the equation which expresses the 

 equality of pressure at all the points of the same level surface. 



As a particular example of the first problem, the figure of equili- 

 brium of a homogeneous fluid is determined on the supposition that 

 it revolves about an axis and that its particles attract one another 

 proportionally to their distance. This example is deserving of at- 

 tention on its own account ; but it is chiefly remarkable, because it 

 would seem at first, from the mutual attraction of the particles, that 

 peculiar artifices of investigation were required to solve it. But in 

 the proposed law of attraction, the mutual action of the particles 

 upon one another is reducible to an attractive force tending to the 

 centre of gravity of the mass of fluid, and proportional to the di- 

 stance from that centre : which brings the forces under the condi- 

 tions of the first problem. 



The second problem investigates the equilibrium of a homogene- 

 ous planet in a fluid state, the mass revolving about an axis, and the 

 particles attracting in the inverse proportion of the square of the 

 distance. The equations for the figure of equilibrium are two ; one 

 deduced from the equal pressure at all the points of the same level 

 surface ; and the other expressing that the stratum of matter be- 

 tween a level surface and the upper surface of the mass, attracts every 

 particle in the level surface in a direction perpendicular to that surface. 

 No point can be proved in a more satisfactory manner than that the 

 second equation is contained in the hypothesis of the problem, and 

 that it is an indispensable condition of the equilibrium. Yet, in all 

 the analytical investigations of this problem, the second equation is 

 neglected, or disappears in the processes used for simplifying the 

 calculation and making it more manageable : which is a remarkable 

 instance of attempting to solve a problem, one of the necessary 

 conditions being omitted. 



The equations found in the second problem, are solved in the third 

 problem, proving that the figure of equilibrium is an ellipsoid. 



3. " Report of a Geometrical Measurement of the Height of the 

 Aurora Borealis above the Earth." By the Rev. James Farquharson, 

 LL.D., F.R.S.* 



The principal object to which the author directed the inquiries of 

 which he here gives an account, is the determination by geometrical 

 nxeasurement of the height of the aurora borealis, and of the altitude 

 and azimuth of the point t9 which the streamers seem to converge, 

 and which has been termed the centre of the corona : these latter de- 

 terminations constituting important data for enabling us to form a 

 clear conception of the whole definite arrangement and progress of 

 the meteor, and also a correct judgement of the degree of reliance 

 to be placed on the methods employed for measuring its height above 

 the earth. The paper is chiefly occupied with the details of the ob- 

 servations made or collected by the author, with their critical dis- 

 cussion, with the correction of some misapprehensions which have 



* A notice of a former paper on this subject, by the same author, will 

 be found in Phil. Mag. and Annals, N.S., vol. vii. p. 355. 



