ORBITS OF LARGE INCLINATIONS AND ECCENTRICITIES. 599 
il ib 1 : j 
Tdi= /1—é{3sinu —-% esin2u— 3sinv + esin (v — w) 
a (22) 
+ esin (v + u) + sin (v — 2u)} es daz) ee 
3 1 
+ {5 e—(8—€) cosu +5 e cos 2u + 3 cosy —3e cos(v—x) 
a dQ 
— ecos(v +u) + cos (v—2u) } ——— a =(7,) 4% 
where it is evident that this quantity has the requisite property. 
From the Fundamenta nova investigationis it follows, that if we 
take account only of the first power of the disturbing force, we 
_ get from T the disturbances of the mean longitude and the cor- 
responding disturbances of the log. of the radius-vector in the 
- following manner. Compute the value of 
we /Tat, 
‘in which integration v must be treated as constant. Hence we get 
nozs=n Wadt; w= a f(F,) a» 
where the stroke over the W and its differential coefficients shows 
that for this integration v must be changed into wv. The con- 
stants to be added to these integrations are here for the sake of 
shortness omitted. In the same work it is shown how to pro- 
_ ceed in the calculation of the disturbances depending on the 
squares, and so forth, of the disturbing force, which method can 
be employed without important change in the problem before 
us. To obtain the disturbances of latitude the author employs 
the elements p, and ¢,, explained in the same work, whose dif- 
ferential expressions are the following :— 
22 ey ie 
ee — cos u) a) 
si = — nacosisinu (a) 
dt a) 7° 
where 7 is the inclination of the orbit uf the comet to any arbi- 
trary fundamental plane. We see that here also the functions 
by which the differential coefficients of © are multiplied possess 
the required condition. After integrating these expressions we 
obtain the disturbance of the latitude 8s by the following ex- 
pression :— 
3s = %q, sin f — dp, cos /; 
