[ 357 3 



LVI. On an Equatmi in Laplace's "Mechanique 



Celeste." 



To Mr. Tilloch. 



JSiR, — JriAViNG observed in the Philosophical Magazine 

 for last Janviarvj a cominunication from Mr. White, re- 

 specting an equation in the MCchanique Celeste of Laplace, 

 I am induced to send you this letter concerning what api- 

 pears to me to be an oversight in the author of that most 

 admirable work. If Mr. White, or any other of your 

 learned correspondents, would favour me with his opinion 

 respecting it, I should esteem it as a particular favour. 

 In vol. i. page 57, of the Mechanique Celeste, Laplace sup- 

 poses that c = S.7n. ^'-fc^^, which he says, page 63 of 

 the same volume, may be put into the following form : 

 II II 



^ „ ■ ' r (x—x)(dy—dt/) — (y—7/){dx—dx)^ -^ 



c.S.mz=S.m7n.^- ^^ ^ ' '^- Now, at 



page 130 of the same volume, he says that the equation 

 const. = S. m. ^^^^^^Lzlffl _ ^-"^ .S.m.^ + ^-"^ 



(//• M+S.m ((/■ il/+2.7J» 



dv 



' ^' *"' If' '^ multiplied by AI + S.m will be changed into 

 the following : 



{xdy-ydx) ' -'' -'•' - •' 



' + X.mm. i (^-■^K-/y-'^j/)-(?/-y)(^^-rfj) T. 



'/y ' I df 



It appears to me from an investigation of these expressions, 



that in the first instance he makes X m, S. m. ^^'^y-V'^). 



II II 



1. ^ ' ( (■'■—x){di/—dij]-(y—yyd.i—dx)-i 



equal to S.mm.^ — ^^^- ^^j , and m the 



second he makes S.m. X.m. - '" ''~^"^'^> — S.m x. S.m.-- + 



<// df 



X.wy. X.m. -jjr equal to the same quantity ; which is im- 

 possible, unless the two last terms of (he last equation be 

 equal to uotlilng; which is not supposed to be the case. 

 I remain, sir. 



Yours respectfully, 



John Thomson, 



Z3 LVI I. Case 



