Laplace's " Micanique Celeste.'* 9 



■with much credit to himself and utility to the public 

 during the last 30 years. The object of the paper is suffi- 

 ciently explained in his short letter prefixed to the calcula- 

 tion. You will oblige us, and 1 doubt not many other of 

 your mathematical readers, by inserting it in the Philoso- 

 phical Magazine. 



Yours very truly, 



James Dinwiddie. 



To Dr. Dinwiddie. 

 Dear Sir, — In the Mtcanique Celeste, vol. i. page 138, 

 Laplace has given an equation marked (B) which is of 

 great use in the theory of the figure of the celestial bodies. 

 To the yonng reader of that profound work the derivation 

 of the above equation may be acceptable j and, if you think 

 that Mr. Tilloch will allow it a place in his Magazine, it is 

 at his service. 



Yours sincerely, 



Dumfries^Mathematical School, ThOMAS WhiTE. 



InLapIace's.equation (^-11) + (^) + (g^) = «, 



marked (A) ; V is a function of x, y, and z; and x is. 

 = r. cos 9j y = r. sin 5. cos tt; and z = r. sin fl. sinTr; and, 



therefore, r- \/x^4-y' + z'; cos Q ;= "" ; and, the 



tang TT = — . Hence, 



( 



—-)= — = cose. 



dx / T 



<f) 



(d6 \ sin tf , • A / 'W \ d. cos C 



•^ ) = - -T J because, - sm e.(--) = -^ 



I X /' dr \ 



r r' ' \ d.v J ' 



(-£ ^ = 0. Therefore, 



(d6 \ sin ^ rf 

 ^; = -7-- 



rf / </*^ \ ,/sin t\ 



\hz'J ? > ^°^» — ^— = :;r- = 



