600 HANSEN ON THE PERTURBATIONS OF BODIES IN 
from whence it follows that 8s cannot be expressed by rapidly 
converging series, since sin f and cos f are not whole functions 
of sin w and cos u; we must then let f be replaced by uw, which 
however the application of the disturbances will render difficult. 
But if the preceding equation be multiplied by 7, then 
ris=59,aV1—e sin u — dp, a (cosu — e) 
in which the required condition is fulfilled. This expression is 
never inconvenient, since it is easy, by the application of the 
disturbances according to their numerical values, to divide this 
by the numerical value of r. But it frequently happens that we 
employ for the computation of the heliocentric places formule 
which require 78 s, and in this case the foregoing expression is the 
most convenient. We can, moreover, by tabulating the disturb- 
ances, employ the preceding expression for 3s, which requires f 
in the place of uw, and make also tables for 8s. 
For carrying out the expansions pointed out in the preceding, 
the differentials to be integrated consist of terms which are 
partly of the form 
nay en b( (¢u+ig' + A) dt, 
cos 
and partly of the form 
SB ere - 
afm hu +ig' + A)du, 
where a and A are independent of ¢ and uw. The former of these 
forms can in two ways be reduced to the latter. We have namely 
in the first place, 
ndt = (1 —ecosu) du, 
and by the substitution of this expression there results 
naf <n (iut+idg! +A)dt=af on (iut+idg' +A) du 
eat an Oh l)ut+iyg'+A)du 
— Seaf n (¢—1)u+i@g' +A)du. 
In the second place we may effect the reduction by integration 
by parts. This gives 
na f sn iutig + A)dt= ar a 
ta ‘COS ;. : 
in yom (@u+ig' +A) du, 
where v = ~. 
ASM iu tig + A) 
