Book Description

Excerpt from Journal of the Institute of Actuaries and Assurance Magazine, 1869, Vol. 14 Hitherto we have only considered the case when the intervals between the given values are equal; but other cases occur in prae tice. For example, in some astronomical observations it happen that the times between the successive observations are unequal and then, taking the time as the variable, the values given by observation are not equidistant, and the problem is to find from the given data the value Of the observed element at any given time. Newton gave the first solution of this problem in a lemma in his Principia, which, according to his custom, he did not stop to demonstrate; and Laplace, who adopted that solution in his method of calculating the orbit of a comet, also confined himself to its simple enunciation. We devote our fifth and last section to the general consideration of cases of this kind; and after giving a solution based on the same purely algebraical reasoning we have used throughout, we shall conclude with an easy demonstration of Newton's solution, under the form given to it by Laplace in his Mécanique Céleste. Lastly, in an Appendix, we have considered more particularly the formulae most frequently used in astronomical interpolations; and we have deduced them also from the same simple principles. We would particularly draw attention to the manner in which we there treat a formula which Stirling, in his excellent work on interpolation, had only obtained by induction, and to the additions we have given to M. Bessel's demonstration of that useful formula. The combined simplicity and clearness of all these calculations will, it is hoped, secure for them a careful perusal. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.