362 CARNEGIE INSTITUTION OF WASHINGTON. 



MATHEMATICAL PHYSICS. 



Moulton, F. R., University of Chicago, Chicago, Illinois. Investigations in 

 cosmogony and celestial mechanics. (For previous reports see Year Books 

 Nos. 4, 5, 8-12.) 



The unpublished investigations of the past year are as follows: 



(1) On the stability of Jupiter's eighth satellite. — G. W. Hill proved, by 

 using the surfaces of zero relative velocity connected with Jacobi's 

 integral, that the moon can never recede beyond a determinate finite dis- 

 tance from the earth. A similar discussion shows that all the remaining 

 known satellites in the solar system, except possibly J VHI, are per- 

 manently attached to their respective primaries. Hill's criterion of sta- 

 bility is not apphcable to J YIII, hence other methods must be used. 



The stability or instability of a periodic orbit has definite meaning 

 capable of mathematical formulation. It is shown that this can be used 

 to determine the nature of the stability of a neighboring non-periodic 

 orbit. In examining the question of the stability of a periodic orbit it 

 is necessary to determine the characteristic exponents of the solutions 

 of certain linear differential equations having periodic coefficients. 

 It is shown how to do this by a relatively simple process which avoids 

 certain lengthy transformations of variables, the expansion of numerous 

 functions in Fourier's series, and the use of infinite determinants. 



A retrograde periodic orbit having the period of the retrograde 

 satellite J VIII has been computed, and the character of its stability has 

 been determined three times by the two methods which have been 

 developed. The orbit is very stable. An equally important question 

 is whether or not a direct satellite having this same period would be 

 stable. After much labor, a direct periodic orbit having the period 

 of J VIII was found. The orbit was unstable. 



This paper was sent in the early summer to the Monthly Notices of 

 the Royal Astronomical Society. 



(2) An extension of the process of successive approximations for the 

 solution of differential equations. — The existing methods of solving dif- 

 ferential equations, with the exception of the Cauchy polygon process, 

 which is not of practical value, have, in general, only a limited domain 

 of applicability. In practical applications, as well as in certain theo- 

 retical discussions, it is important to have means which will furnish 

 the solution in any arbitrary part of the domain of its existence. The 

 processes defined in this paper have this advantage. As an application, 

 they make it possible to lay down a complete logical foundation for the 

 so-called method of mechanical quadratures, which has been extensively 

 employed by Hill and Darwin in discovering periodic orbits of the 

 problem of three bodies. 



(3) Orbits of ejection from one body and collision with another. — Closed 

 orbits of ejection (that is, those which are orbits of ejection from a body 

 and of collision with the same body) were treated in a paper which was 

 published in the Proceedings of the London Mathematical Society, 



