LAPLACE. 475 



887. The present memoir may be regarded as a collection of 

 examples in the theory of Finite Differences ; the methods ex- 

 emplified have however since been superseded by that of Gene- 

 rating Functions, which again may be considered to have now 

 given w^ay to the Calculus of Operations. The problems involve 

 only questions in direct probability ; none of them involve what 

 are called questions in inverse probability, that is, questions 

 respecting the probability of causes as deduced from observed 

 events. 



888. In the same volume as the memoir we have just ana- 

 lysed there is a memoir by Laplace entitled, Menioire sur Tincli' 

 naison moyenne des orbites des cometes ; sur la figure de la Terre, 

 et sur les Fonctions. The part of the memoir devoted to the mean 

 inclination of the orbits of comets occupies pages 503 — 524 of the 

 volume. 



In these pages Laplace discusses the problem which was started 

 by Daniel Bernoulli ; see Art. 395. Laplace's result agrees wdth 

 that which he afterwards obtained in the Theorie...des Proh. 

 pages 253 — 260, but the method is quite different ; both methods 

 are extremely laborious. 



Laplace gives a numerical example ; he finds that supposing 

 12 comets or planets the chance is "339 that the mean inclination 

 of the planes of the orbits to a fixed plane will lie betw^een 

 45° — 7^" and 45", and of course the chance is the same that the 

 mean inclination wiJl lie between 45"^ and 45^ + 7^°. 



889. The volume with which w^e have been eno^aofed in Arti- 

 cles 881 — 888 is remarkable in connexion with Physical Astronomy. 

 Historians of this subject usually record its triumphs, but omit its 

 temporary failures. In the present volume Lagrange affects to 

 shew that the secular acceleration of the Moon's motion cannot be 

 explained by the ordinary theory of gravitation ; and Laplace 

 affects to shew that the inequalities in the motions of Jupiter and 

 Saturn cannot be attributed to the mutual action of these planets : 

 see pages 47, 213 of the volume. Laplace lived to correct both his' 

 rival's error and his own, by two of his greatest contributions to 

 Physical Astronomy. 



