488 LAPLACE. 



angle instead of one right angle whicli he takes in the Tlieorle... 

 des Prob. Laplace's words are, on page 362 of the memoir : 



Si Ton fait varier les inclinaisons depuis zero jiisqu'a la demi-cir- 

 conference, on fait disparoitre la consideration des mouvemens retro- 

 grades j car le mouvement direct se change ^en retrograde, quand I'incli- 

 naison surpasse un angle droit. 



Laplace obtains in the memoir the same numerical result as on 

 page 258 of the Theorie...des Proh. ; but in the latter place the 

 fact of the motions being all in the same direction is expressly 

 used, while in the former place Laplace implies that this fact still 

 remains to be considered. 



The calculation for the comets, which follows some investiga- 

 tions noticed in the next Article, does not materially differ from 

 the corresponding calculation in the Theorie . . .des Proh.; 97 is 

 taken as the number of comets in the memoir, and 100 in the 

 Theorie . . . des Proh. 



916. Laplace gives an investigation the object of which is 

 the approximate calculation of a formula which occurs in the 

 solution of the problem noticed in the preceding Article. The 

 formula is the series for Zl'^s', so far as the terms consist of 

 positive quantities raised to the power which i denotes. A large 

 part of the memoir bears on this subject, which is also treated 

 very fully in the Theorie... des Proh. pages 165 — 171, 475 — 482. 

 This memoir contains much that is not reproduced in the 

 Theorie... des Proh., being in fact superseded by better methods. 



We may remark that Laplace gives two methods for finding the 



value of I fe~'^^'' cos htdt, but he does not notice the simplest 



'' 



method, which would be to differentiate e-^^^ cos htdt four times 



-J 



with respect to h, or twice with respect to c ; see pages 368 — 370 

 of the memoir. 



917. In pages 383 — 389 of the memoir we have an important 

 investigation resembling that given in pages 329 — 332 of the 

 TJieorie...des Proh., which amounts to finding the probability that 

 a linear function of a large number of errors shall have a certain 



