the Tores of Saturn. 475 
out ; the greater number falling in until at last they are 
brought down upon the body of the planet. 
Now the interest of the observations at Flagstaff consists 
in their showing us this disintegration of the rings in process 
of taking place, and furthermore in a way that brings before 
us an interesting case of celestial mechanics. 
In considering the action of one body upon another revolving 
around a third, the points germane to our present inquiry 
are the perturbations in the radius vector and in the longitude 
of the second body. 
Now, by the method of variation of parameters the radius 
vector of the perturbed body — the disturbed particle in the 
present case— may be expressed, as has been done by Airy, by 
ri = 
%(l-*i 2 ) 
1 + ^cos (#!— ft>i)' 
where the subscripts refer to the variable elements. The 
perturbed longitude may similarly be expressed by 
6 l = n\t + e l + ( 2e ± — -jr- + &c. ) sin [n ± t + e x — a){) + 
&c. 
These may be expanded in an ascending series of terms 
according to powers of the excentricities and of cosines 
of multiple arcs of the mean motions of perturber and per- 
turbed by Fourier's series. The resulting expression is 
composed of terms similar to those in the undisturbed orbit, 
and of others denoting the effect of the perturbation. The 
latter are of the typical form : 
PeV 
pn—qn 
,cos (pn — qn^t — Q, 
where P is a function of a and d , the radii vectores of the 
perturber and perturbed, and of jju the mass of Saturn and 
the perturbed body. 
The form of these terms shows that they will become con- 
siderable in proportion as pn—qn' is small, since their 
coefficients are divided by this quantity. Now as n and ri 
are the mean motions of perturber and perturbed, if these 
are commensurate there will always be terms of the sort 
which will be large, namely, those in which - = - ; for v and a 
p n r * 
are always integers, in consequence of the method of 
expansion. 
