Page:Lectures on Ten British Physicists of the Nineteenth Century.djvu/134

 observed in 1866, and it seemed probable that the meteors and the comet constituted one moving aggregation. In 1899, thirty-three years later, an exceptional display of meteors was predicted on the strength of Adams' result; there was much popular lecturing on the subject beforehand; the citizens of London on the predicted night went to bed having previously arranged with the policeman on the beat to call them up, but their slumbers were not disturbed.

Eleven years later (1877) Adams recognized the merits of an American astronomer George W. Hill, who was then an assistant in the office of the American Nautical Almanac, and whose eminence as an astronomer is now universally recognized in the world of science. Hill in 1877 published a paper on the motion of the moon's node in the case when the orbits of the Sun and Moon are supposed to have no eccentricities, and when their mutual inclination is supposed to be definitely small. He made the solution of the differential equations depend on the solution of a single linear differential equation of the second order which is of a very simple form. This equation is equivalent to an infinite number of algebraical linear equations, and Hill showed how to develop the infinite determinant corresponding to these equations in a series of powers and products of the small quantities forming their coefficients. Adams in his unpublished investigations had discovered the same infinite determinant, and was thus in a position to immediately recognize the value of Hill's work. This same year (1877) Adams communicated to the British Association at Plymouth the results of a calculation of Bernoulli's numbers. Bernoulli's numbers are the coefficients of $$x^n/n!$$ in the expansion of $$x/(1-e^{-x})$$. Now $$\frac{x}{(1-e^{-x})}=1+\frac{1}{2} x +B_1\frac{x^2}{2!} - B_2\frac{x^4}{4!}+B_3\frac{x^6}{6!} -\ldots$$ in which $$B_1=\frac{1}{6}, B_2=\frac{1}{30},\ldots B_9=\frac{43867}{798}$$ The first fifteen $$B's$$ were calculated by Euler, the next 16 by Rothe; and in this communication Adams supplied the following 31 numbers. The difficulty of this calculation may be judged from the facts. that the denominator of $$B_32$$ is 510 and the numerator is a