SCIENCE 



[N. S. Vol. XXXIII. No. 838 



of the gap at the ratio 2 : 1, this being the 

 most extended of all. Here there are no 

 known planets with periods between 2,120 

 and 2,204 days ; the exact ratio is 2,166, not 

 very far from half way between the two 

 sides of the gap. The comparatively few 

 planets on the outer side of the gap may be 

 a consequence of the general distribution, 

 but the sudden increase in the number on 

 the inner side suggests that other causes 

 have been at work. 



In attempting to give an explanation of 

 the phenomena by gravitational forces 

 alone, I must on this occasion omit any 

 mathematical discussion and simply lay be- 

 fore you the main results at which I have 

 arrived. Until this discussion is published 

 in detail, you will naturally reserve your 

 opinions as to the validity of the argument. 

 The reasons for presenting the results in 

 advance of the methods will be evident in 

 the course of my remarks. 



The motion of an asteroid near the crit- 

 ical places depends mainly on two quanti- 

 ties, the apparent eccentricity of the planet 

 at any time and the difference between the 

 actual period and the period which is twice 

 that of Jupiter. In the theory both these 

 are quantities such that if at any moment 

 the attraction of Jupiter were suddenly 

 annihilated, the asteroid would continue 

 its motion in an elliptic orbit round the 

 sun with this period and eccentricity. 

 They are thus determined by the position 

 and velocity of the planet at the moment. 

 The attraction of Jupiter causes the tem- 

 porary period and eccentricity to vary and 

 it is the variations of those two quantities 

 that are the main factors. Now the equa- 

 tions which I have obtained give the prin- 

 cipal parts of these two quantities in the 

 form of the square roots of variable func- 

 tions. In order that they may be real 

 these variable functions must always be 

 positive. One of two things must happen. 



Either the variability of the functions is 

 forced, that is, it is independent of the mo- 

 tion of the asteroid, or it is free. In the 

 latter case the motion of the asteroid is 

 limited. Under certain general conditions 

 it has long been known that the equation 

 giving the period is to a large extent free 

 so that the asteroid may librate or not, ac- 

 cording to the values of the constants en- 

 tering into the equation, and there is noth- 

 ing to prevent the constants from having 

 such values. But the period also depends 

 on the eccentricity, and the possible varia- 

 tions of this quantity can not be neglected. 



The eccentricity and the difference of 

 the actual period from that of Jupiter are 

 really connected by two equations which 

 can not be independently treated. The 

 eccentricity is also dependent on the short 

 period terms which from this point of view 

 may be considered as of forced period, 

 since the periods of the larger terms are 

 very nearly multiples of the period of 

 Jupiter. I find from this that the eccen- 

 tricity can not permanently remain below 

 a certain limit and consequently that if a 

 libration of the period about the critical 

 ratio occurs, it can not have a very small 

 amplitude. Eeferring to the analogy of 

 the rod, we discover the presence of forces 

 which prevent the angle of oscillation of 

 the rod to and fro from descending below 

 a certain finite value. The oscillation can 

 not become infinitely small. This result is 

 in agreement with a theorem proved by 

 Poincare that no periodic orbit at such a 

 critical place exists. 



Beducing to numbers, I find that the 

 lower limit of the slow variation of the ec- 

 centricity for an asteroid just before it 

 reaches the critical stage between libration 

 and non-libration is about one twentieth 

 and that at some time it must become as 

 large as one seventh. If there is an aster- 

 oid which has passed this stage and is 



