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XLIV. On Laplace's Equation. By John H. Jellett, 

 D.D.j Provost of Trinity College, Dublin*. 



THE equation upon which Laplace founded his discussion 

 of the problem of attraction has excited, as is well known, 

 a large amount of controversy. No doubt, indeed, appears to 

 exist as to the truth of this equation in the case to which alone 

 Laplace has applied it, nor therefore as to the validity of the 

 method which he has built upon this foundation. But Laplace 

 professes to demonstrate the truth of his equation for a case 

 more general than that which he subsequently considers: and 

 it is in this more general form that its truth has been ques- 

 tioned. My object in the present paper is to endeavour to 

 determine exactly the limits within which Laplace's equation 

 is true, as it appears to me that these limits are somewhat 

 wider than has been generally supposed! . 



I proceed, in the first place, to give a proof of this equation 

 agreeing substantially with that given by Laplace, and then to 

 consider what conditions or limitations, if any, are necessary to 

 the validity of this proof. Laplace's equation is as follows : — 



Suppose the force of attraction to vary as the nth power of 

 the distance. Let Y be the potential of the attraction of a 

 nearly spherical body on a point upon its surface. Let Y 5 be 

 the potential (at the same point) of a sphere, touching the sur- 

 face at the point in question, and separated from it throughout 

 by a small distance, which we may denote by au, a being a 

 small constant quantity, and u a function of the polar angles 

 6, cf). Let A, A 1 be the attractions, resolved along the common 

 normal, of the original solid and the sphere respectively. Let 

 also b be the radius of the sphere. Laplace's equation is 

 (M4c. CM. Livr. 3, chap. ii. sect. 10), 



9h 



v = v 1 +^Ti( A - A .) « 



This equation may be put under a form somewhat simpler, 

 and which will enable us to mark more distinctly the limits 

 within which it is true. Let v be the potential, at the point 

 of contact, of the shell included between the original surface 

 and the sphere. Let also a be its normal attraction at the 

 same point. Then 



v=Y— Y l3 a = A—A 1 , 



* Communicated by the Author. 



t " It may "be taken, I think, as universally admitted that the equation 

 cannot be considered established if n is negative and numerically greater 

 than 2 " (Todhuntcr, ' History of the Mathematical Theories of Attrac- 

 tion and the Figure of the Earth,' vol. ii. p. 263). 



