TITKE GENERAL PERTURBATIONS OF TILE MINOR PLANETS. 9 
given with some detail. And while the writer is fully aware that here he may have 
exposed himself to criticism, it will suffice to state that he has not had in mind those 
competent of doing better, but rather the large class of persons that seems to have 
been deterred thus far, by imposing and formidable-looking formulz, from becoming 
acquainted with the means and methods of theoretical astronomy. In the present 
state of the science there is greatly needed a large body of computers and investiga- 
tors, so as to secure a fair degree of mastery over the constantly growing material. 
The numerical example presented with the theory for the purpose of illustrating 
the new method will be found to cover a large part of the treatise. The example is 
designed to make evident the main steps and stages of the work, especially where 
these are left in any obscurity by the formule themselves. As a rule, the formule are 
given immediately in connection with their application and not merely by reference. 
It has been the wish to make this part of the treatise helpful to all who desire to 
exercise themselves in this field, and especially to those who desire to equip themselves 
for performing similar work. 
The time required to determine the perturbations of a planet according to the 
method here given is believed to be very much less than that required by the unmodi- 
fied method of Hansen. Nearly all the time consumed in making the transforma- 
tions by his mode of proceeding is here saved. The coefficients b are much more 
quickly and readily found by making use of the tables prepared by RuNKLE, giving 
the values of these quantities. Doubtless experience will suggest still shorter pro- 
cesses than some of those here given and thus bring the subject within narrower limits 
in respect to the time required. If we compare the time demanded for the computa- 
tion of the perturbations of the first order, with respect to the mass, produced by 
Jupiter, with the time needed to correct the elements after a dozen or more oppositions 
of the planet, computing three theoretical positions for each opposition, it is believed 
there will not be much difference, if any, in favor of the latter. 
Again, when we wish to find only the perturbations of the first order, experience 
will show where many abridgments may safely be made. And whenever the positions 
of these bodies are made to depend upon those of comparison stars whose places are 
often not well determined, it will be found that the quality of the observed data 
does not justify refinements of calculation. 
One of the things most needed in the theory of the motions of the minor planets 
is a general analytical expression for the perturbing function which may be applicable 
to all these small bodies. Thus if we had given the value of aQ in terms of a periodic 
series, with literal coefficients and with the mean anomalies of the planets as the argu- 
2X5 JES Sk A\VOlke IIDG 18. 
