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Prof. P. Lowell on the Asteroid*. 347 



From the results of these tables we perceive that only 



three of the asteroids in the n — 2n' gap, to wit (122)' uos)' 



and (W), show libration. Whereas were the asteroids as 



uniformly distributed throughout the n = 2n gap as they 

 are on both sides of it, we ought to find 40 or more where 

 we now find none. As the gaps throughout the whole 

 asteroid zone occur where and only where their periods are 

 commensurable with that of Jupiter — up to those involving 

 the seventh power of the eccentricities — we see that the 

 arrangement cannot be fortuitous, and furthermore from 

 the present investigation that permanent not libratory action 

 is necessary to account for it. 



12. We now take up the case of the gap n=3n'. For 

 this R becomes for the terms in 6 y 6\ and 9", all of which 

 are reducible to #, 



[e\2lb® + 10 b^ + b 2 &) - ee (40 b® -f 20 b^ + 2 bp) ) cos (*r - «■') J cos 

 - [ee' (406< 3 > + 20^ 2 > + 2 b 2 &) sin («■ - «r'] sin 6 

 + [e' s [17(6-« -a) + 10 (& x « -a) + ^ 2 (1) ] cos2(w— B 7 ) ] cos# 



+ [ 6 /2 [17(// 1 )-a) + 10(/V 1) -«) + ^ (1) ]sin2(^- OT ']sin6>T 



Letting 7i' = e 2 [21Z/ 3 > +10 ^( 3 > +^ 2 (3) ], 



and A" == ee' [40 M 2 > + 20 b^ + 26 2 < 2 > ] 



7 t "W 3 L17(&< 1 >-a)4^0(&< 1 >-a) + W 1 >, 



we get in this case, by a process similar to that for n — 2n\ 



j a 



~- = V - i A(C0S (6 -ry) + C), 



in which 



A= ^7 V[{/*' *-/*'' cos (<*-<*') + A" / cos2(>-sr'0} 2 

 + {A 7 sin ('57 — -53-') — /t"' sin 2^ — sr')}-]. 



This equation in -7- can be put in the form 



(9) ~= VAsin i(^- 7 ) or = yAeosJ(^7)f 



according to the sign of A. 



