208 Mv. S. Tebay on the Law of Bode; 



I have not succeeded in obtaining a satisfactory solution of 

 this problem^ so long as the third planet is taken into account ; 

 but on considering the motion of a planet disturbed by an interior 

 one, I have fallen (in my lucubrations) on a rather singular co- 

 incidence, which I now proceed to describe. 



Let a be the semi-major axis of the elliptic orbit of a disturbed 

 planet {m), «' the semi-major axis of the elliptic orbit of the dis- 

 turbing planet (»i') . Then, neglecting the excentricities, we have 



Br= 77 m'a^ — ? 1- &c., 



6 da 



where 



Now it is remarkable that the expression 



„3rfA»> 



-^{Mm'^iim--'-}. 



da 

 admits of a minimum value, viz. when 



■<mh^-Q^my 



+ &c.=l. 



Perhaps the readiest way of determining - from this equation, 



is by the rule of double position. Let m„ be the general term of 



a 

 the series; then putting -=«, we have 



a 





M„+, _27l-f3/2n+_l_l^^^)2. 



u„ 



2«-l L2(?i-M). 



/. log ii„+ 1 = log Ua + 2 log 2n -f 1 -f log 2n + 3 

 -2 log M^ - log 2n-l -2 log 2 

 + 21og(«). 

 Suppose a =|. Then 



«i = T6^ log Ml = 1-2730013 J 

 and the other terms are calculated from the formula 

 logM«+i = logM„ + 21og2w4-l + log2« + 3 

 -21ogra4-l-log2n-l-41og2. 



