43 !• Mr. Ivory cni the Theory of the Figure of the Planets 



account of its thinness, it may be considered as a series of 

 molecules spread over the surface of the sphere. Conceive 

 the surface of the sphere to be divided into an infinite number 

 of elementary parts, or differentials, as d s; the position of 

 d s being determined by the arcs 9' and ct', or \I/ and 4> : then 

 the distance of ds from the assumed point will be equal to 

 \/ )^ — 2r a cos ^ -{- a^ ; and the thickness of the molecule 

 standing upon d s being « a y, the part of V relative to the 

 whole stratum will be equal to ^ , ^ 



Aj.^ — 2 r a cos \J/ + a^) ^ . « ay. d s, 



the integral being extended to the whole surface of the sphere. 

 If now we use f to denote \/ r~ — 2 r a cos 4* + ^^ and add 

 together the two parts of V, we shall get 



V = A +y/""*''.«ay.rf5. 



Differentiate this equation with respect to r, theii 



r — = r- — h [n + I) f{r'^ — r a cos ■V)f^~ • « «y« ^^- 



But, r'^ — r a cos i^ = ^f^ + \ (r^ — a-); wherefore 



d V d A . n -t- 1 rr^ 4-1 17 'i + l/Q 9\ /\'W — 1 j 7 



'■ dT == ""d^ + 2/^ .uaj/.ds + -^ {r-'--a^)fj . « ay. rfs. 



If now we combine this expression with the value of V, we 

 shall readily obtain 



—I— V — r — = — !— A — r ^!— (r- — a^)/f .a. a y. d s. 



Here the last term on the right hand has the factor r^ — a% 

 which is very small, because a is nearly equal to r : and if we 

 suppose that a increases till the sphere touches the spheroid, 

 the factor will be rigorously evanescent, and the term multi- 

 plied by it will, generally speaking, likewise vanish. Now if 

 we observe that when a increases to be equal to r, the thick- 

 ness uat/ becomes « a (y' — y) the last equation may be se- 

 parated into these two formulae, which, genei'ally speaking, are 

 true when r = a, viz. 



n-\-\^j d V n + 1 . d A ~1 



{r'''-a')ff'-\y' -y)ds = j 



The first of these formula? is Laplace's equation at the sui'face 

 of the spheroid ; and it is true in every case when the inte- 

 gral in the second, after being multiplied by the evanescent 

 factor ?-^ — a^, makes a product equal to zero. Having now 

 brought the matter under consideration to a form proper for 



