8 A NEW METHOD OF DETERMINING 
From the value of IW we derive that of W by simple mechanical processes, and 
then the perturbations of the mean anomaly and of the radius-vector are found from 
pe Oe = nf W .dt 
y being a particular form for g. 
The perturbation of the latitude is given by integrating the equation 
C being a factor found in the same manner that A and B were. 
It will be noticed that in finding the value of . dz two integrations are needed ; 
in finding the perturbation of the latitude only one is required. 
The arbitrary constants introduced by these integrations are so determined that 
the perturbations become zero for the epoch of the elements. 
In all the applications of the method of this paper to different planets the cireum- 
ference has been divided into sixteen parts, and the convergence of the different series 
is all that can be desired. In computing the perturbations of those of the minor 
planets whose eccentricities and inclinations are quite large, it may be necessary to 
divide the circumference into a larger number of parts. In exceptional cases, such as 
for Pallas, it may be necessary to divide the circumference into thirty-two part s. 
In the different chapters of this paper the writer has given all that he conceives 
necessary for a full understanding of all the processes as they are in turn applied 
And he thinks there is nothing in the method here presented to deter any one with 
fair mathematical equipment from obtaining a clear idea of the means by which astron- 
omers have been enabled to attain to their present knowledge of the motions of the 
heavenly bodies. The object always kept in mind has been to have at hand, in conve- 
nient form for reference and for application, the whole subject as it has been treated by 
HANSEN and others. Thus in connection with HANseEwn’s derivation of the function 
IV, to obtain clearer conceptions of some matters presented, the method of BRuNNow 
for obtaining the same function has also been given. In some stages of the work 
where the experience of the writer has shown the need of particular care the work is 
