ORBITS OF LARGE INCLINATIONS AND ECCENTRICITIES. 597 
According to the foregoing division of the subject, it is now to 
be shown how we are to proceed in the second case, in which 
ris>r. 
On account of the small eccentricity of the disturbing planets 
it is not necessary, at least in the greater number of cases, to 
avoid infinite series proceeding according to the powers of the 
eccentricity of the planet. We certainly lose thereby somewhat 
of the natural convergence of the disturbance-function, but the 
diminution which it receives is not so great as to be hurtful; on 
the other hand, while something is lost in this respect, we gain 
an advantage with relation to the facility of the integration and 
the subsequent application of the disturbances. Instead of the 
form just given for the expansion of the disturbance-function, 
the author employs, for the case now treated of, the follow- 
ing :— 
OX =>M; i cos tu + 7g’) + 2N, i sin (iu + ig), 
where g’ is the mean anomaly of the disturbing planet. 
It is now a matter of indifference, for the end here to be at- 
tained, which method is employed for carrying out the expan- 
sion of © and its differential coefficients, provided only that by 
expansion the preceding formula be adapted for use, and the 
values of the coefficients be fully obtained; for to this form cor- 
respond determinate values of the coefficients M; ; and N; ,, and 
therefore the other method of development, if only it be based on 
correct principles, and be capable of complete development, must 
lead to the same values of these coefficients. Itmay be most advan- 
tageous in one particular case to employ the latter, and in others 
to employ the former method. In the calculation of the disturb- 
ances of Encke’s comet by Saturn, before exhibited, the author 
has employed for the expansion of the differential coefficients of 0 
the same analysis and the same quantities which were useful in 
the foregoing, for the purpose of finding the form which must be 
given to the expansion in the problem before us to produce the 
greatest possible amount of convergence. The latter method 
has in this instance led very rapidly to the desired result, for the 
expansion of the differential coefficients of O required only the 
labour of two days. The separation of the formule necessary 
for this method must, for the sake of brevity, be here omitted, 
and we proceed therefore with the explanation of the remaining 
part of the memoir. 
The author employs for the calculation of the disturbances in 
