It is difficult to convey to one who is not conversant with such investigations, an adequate notion of the labor which such an inquiry involved. The calculation of the influence of any one _planet_ of the system upon any other is itself a problem of some complexity and difficulty; but still, one general computation, depending upon the calculation of the terms of a certain series, is sufficient for its solution. This comparative simplicity arises entirely from two circumstances which characterize the planetary orbits. These are, that, though they are ellipses, they differ very slightly from circles; and though the planets do not move in the plane of the ecliptic, yet none of them deviate considerably from that plane. But these characters do not belong to the orbits of comets, which, on the contrary, are highly eccentric, and make all possible angles with the ecliptic. The consequence of this is, that the calculation of the disturbances produced in the cometary orbits by the action of the planets must be conducted not like the planets, in one general calculation applicable to the whole orbits, but in a vast number of separate calculations; in which the orbit is considered, as it were, bit by bit, each bit requiring a calculation similar to the whole orbit of the planet. Now, when it is considered that the period of Halley's comet is about seventy-five years, and that every portion of its course, for two successive periods, was necessary to be calculated separately in this way, some notion may be formed of the labor encountered by Lalande and Madame Lepaute. "During six months," says Lalande, "we calculated from morning till night, sometimes even at meals; the consequence of which was, that I contracted an illness which changed my constitution for the remainder of my life. The assistance rendered by Madame Lepaute was such, that, without her, we never could have dared to undertake this enormous labor, in which it was necessary to calculate the distance of each of the two planets, Jupiter and Saturn, from the comet, and their attraction upon that body, separately, for every successive degree, and for one hundred and fifty years."