476 Prof. P. Lowell on 
The various terms with the argument (pn — qn')t will have 
coefficients of different powers of the excentricities. The 
lowest of these which can occur in the expressions will be of 
the order p — q* The term, therefore, in which #4 x x =p — q 
is the term least diminished by the excentricity coefficient, 
and therefore the most potent in its effect. 
From this it is evident that two bodies will mutually disturb 
each other in their revolutions about a third according as 
their periods are : 
1st. Commensurate. 
2nd. Differ bv the smallest integer. 
The most disturbing ratio is when the periods are: — 
1:2; 2 : 3, &c. ; 
the next, 1:3; 3:5, &c. ; 
then, 1:4; 2 : 5, &c. ; 
and so on. 
The initial ratio in each line will be the most effective in 
that line, because the cycle of the disturbance will be repeated 
in the time it takes the outer body to come again into con- 
junction with the inner, and this for the ratio 3 : 5, for 
instance, will be three times as long as for that of 1 : 3. 
The same thing can be seen geometrically by considering 
that the two bodies have their greatest perturbing effect on 
one another when in conjunction, and that if the periods of 
the two be commensurate, they will come to conjunction over 
and over again in the same points of the orbit, and thus the 
disturbance produced by one on the other be cumulative. If 
the periods are not commensurate the conjunctions will take 
place in ever shifting positions, and a certain compensation 
be effected in the outstanding results. In proportion as the 
ratio of periods is simple will the perturbations be potent. 
Thus with the ratio 1 : 2 the two bodies will approach closest 
only at one spot, and always there, until the perturbations 
thus induced themselves destroy the commensurability of 
period. With 1 : 3 they will approach at two different spots 
recurrently ; with 1 : 4 at three, and so on. The number of 
points round the orbit at which they will meet is in fact as 
the sum of the powers of the excentricities in the lowest 
coefficient of the terms with the commensurable argument. 
We see, then, that perturbations, which in this case will 
result in collisions, must be greatest on the particles having 
periods commensurate with those of the satellites. But 
