24. 



ON THE MOTION OF THE MOON'S NODE IN THE CASE WHEN THE 

 ORBITS OF THE SUN AND MOON ARE SUPPOSED TO HAVE NO 

 ECCENTRICITIES, AND WHEN THEIR MUTUAL INCLINATION IS SUP- 

 POSED TO BE INDEFINITELY SMALL. 



[From the Monthly Notices of the Royal Astronomical Society. Vol. xxxvin. (1877).] 



A VERY able paper has recently been published by Mr G. W. Hill, 

 assistant in the office of the American Nautical Almanac, on the part of 

 the motion of the lunar perigee which is a function of the mean motions 

 of the Sun and Moon. 



Assuming that the values of the Moon's coordinates in the case of no 

 eccentricities are already known, the author finds the differential equations 

 which determine the inequalities which involve the first power of the eccen- 

 tricity of the Moon's orbit, and, by a most ingenious and skilful process, 

 he makes the solution of those differential equations depend on the solution 

 of a single linear differential equation of the second order, which is of a 

 very simple form. This equation is equivalent to an infinite number of 

 algebraical linear equations, and the author, by a most elegant method, 

 shews how to develop the infinite determinant corresponding to these 

 equations in a series of powers and products of the small quantities forming 

 their coefficients. The value of the multiplier of each of such powers and 

 products as are required is obtained in a finite form. By equating this 

 determinant to zero, an equation is obtained which gives directly, and 

 without the need of successive approximations, the motion of the Moon 

 from the perigee during half of a synodic month. The small quantities 



