138 SEE— DYNAMICAL THEORY [April 19, 



treating of this at length, we shall recall a suggestive investigation of 

 Sir William Herschel printed in the Philosophical Transactions for 

 1802 (pp. 477-502) under the title " Catalogue of 500 New Nebulze, 

 Nebulous Stars, Planetary Nebulse, and Clusters of Stars; with Re- 

 marks on the Construction of the Heavens." 



VII. Herschel's Theorem on the Motion of Multiple Stars, 



1802. 



In the important paper just cited Herschel first discusses 



" Binary Sidereal Systems or Double Stars," and then proceeds to 



Section " III. Of more complicated sidereal systems, or treble, cjuad- 



ruple, quintuple and multiple stars," where he reasons as follows : 



" In all cases where stars are supposed to move round an empty center, in 

 equal periodical time, it may be proved that an imaginary attractive force may 

 be supposed to be lodged in that center, which increases in a direct ratio of 

 the distances. For since, in different circles, by the law of centripetal forces, 

 the squares of the periodical times are as the radii divided by the central 

 attractive forces, it follows, that when these periodical times are equal, the 

 forces will be as the radii. Hence we conclude, that in any system of bodies, 

 where the attractive forces of all the rest upon any one of them, when reduced 

 to a direction as coming from the empty center, can be shown to be in a 

 direct ratio of the distance of that body from the center, the system may 

 revolve together without perturbation, and remain permanently connected 

 without a central body." 



This reasoning is best understood by means of simple formulae: 

 Let /i and /o be two centrifugal forces, which in revolving systems 

 are always equal to the centripetal forces, and F^ and Vo the cor- 

 responding velocities of the bodies, and i\ and To the radii of the 

 circles in which they are supposed to revolve. Then, by the ele- 

 mentary principles of mechanics, we have 



fi= ,. ; A = -- ; whence/, == -yj— ; /, = -.,— - . 

 1 2 11 22 



This gives 



/2 4'^'^'i. .2_4tV2 .^,N 



A =— ^, ^2 =-y-— • (41) 



Now in orbital revolution the centripetal and centrifugal motions 



