126 SEE— DYNAMICAL THEORY [April 19, 



to the sphere or znce versa. Therefore both U and [' may become 

 infinite from the appulses of stars under the secular effects of the 

 mutual gravitation of the stars of the cluster. If we denote by y^C7 

 and y-F the Laplacean operation indicated in (16), as applied to the 

 functions U and V, the right members of equations (19) and (20) 

 when modified to include the appulses accumulating in a cluster over 

 long ages, become 



J J U^-^JS - J J f U{^^ V)dxdydz - E ^-rrUl 



^ // ^ ^'^i'^^ ~ Iff ^^^"^y^'^y^' - |j 4^ VI. 



In our present problem the triple integrals may be neglected, since 

 y^F and -^-U are each zero, or evanescent, in the small spheres 

 where the appulses occur, and even here are small quantities of the 

 order a-. Hence by (14) the secular equations become 



r. rr r/dudv eudv dUdv\ , , , 





dU ^ 



V^-dS-T.^A-r^V: (22) 



U^^dS-Y.^^'rrW. (23) 



Over very great intervals of time, to be reckoned, as Herschel 

 believed, in " millions of ages," the number of appulses may be taken 

 to be proportional to the time in either the original shell or the 

 original sphere. Consequently instead of the summations in the 

 right members of (22) and (23) we could introduce terms depending 

 directly on the time, and thus write 



= j J U^^dS-A'rrU'-^it-Q 



