594 HANSEN ON THE PERTURBATIONS OF BODIES IN 
and H the cosine of the angle which the radii-vectores include. 
These two series do not converge in all cases, for if 7 be > 7’, the — 
first series never converges, and if 7 be <7’, the second series 
never converges; for this reason, in the problem in question, 
the two cases corresponding to 7 <7’, and r > 7’, must be taken 
separately. In the first case the first series is always conver- 
gent, and in the latter case the second is always convergent. If 
r = 7, both series converge, with the exception of the case where 
at the same time H = + 1. But this case supposes the coming 
together of the comet and planet, in which generally the com- 
putation of the disturbances ceases to be possible. 
The author names the degree of convergence which the above 
series present from one term to another, when they are expanded 
according to cosines of multiples of the angle whose cosine is H, — 
the natural convergence of the disturbance-function, and it is 
his object that this shall be by no means increased, but on the 
contrary be diminished by means of further development. In 
this further development it is necessary to set out with the sup- 
position that the natural convergence of the disturbance-function 
and of its differential coefficients is possible. 
The integrals 
ore RES | 
d ae 
1 J A eh A 
Au, 4. + ss%gieis sok ac ae 
2 aiteks ; 3 
converge more rapidly than the quantity - c itself and its diffe- 
rential coefficients. This law sometimes suffers an exception, 
which, however, concerns only individual terms, which will be 
increased by integration. Precisely in those cases where the 
natural convergence of the differentials is least, it will commonly 
be most increased through integration. 
Through the expansion of the disturbance-function in multi- 
ples of the sine and cosine of the mean anomalies of both the 
bodies in question, and the thereby necessary and inevitable ex- 
pansion of the coefficients in infinite series proceeding according 
to powers of the eccentricity and inclination, whether we repre- 
sent them explicitly or express their sums, that is, the coefficients 
themselves, by means of transcendents, the natural convergence 
of the disturbance-function, even when the eccentricity and 
inclination are small, is remarkably diminished; and when 
these quantities are of considerable magnitude, the natural con- 
