282 Mr. MacCullagh on a Difficulty in the. Theory 

 - 1— cosp,<x)p f/ + 2sin p, sinp„ cot 2 A tan 2 j p— p u 



? 



sin 9 \ ftcsa p^ + sin p { sin p y/ cot 2 X tan 2 | p— p u 



sin 2 p 



m tanlif,^ J cos2 in9 og5 i 



sin a Asin 2 p (. n ru j 



m ^f^M{ l-«n«xsin*i^ „}••* (25) 



sin 2 Asin 2 p (. 2n r "J 



Substituting in (25) the values of p adapted to the two cases 

 of P/Cv>p y/ and pj-\-p ifl we shall have expressions for the corre- 

 sponding times. It is altogether unnecessary to consider them 

 here, or to enter upon a discussion of any other analogies be- 

 tween the properties, which Delambre has collected together, 

 of the diagram employed, and those which belong to the more 

 general problem discussed in this paper; since they do not all 

 hold true, and where they do hold true they are mere matters 

 of mathematical curiosity, — at the same time that they offer no 

 real difficulties to any one who may be tempted to enter upon 

 their investigation. 



Bath, August 8, 1833. 



* # # Mr. Skene, in the prefatory remarks to his solution, 

 mentions an analytical one by Fontana ; but he does not say 

 whether it was a separate publication or inserted in any of 

 the foreign scientific Transactions. I have sought it without 

 success. A solution by means of the Stereographic Projection 

 of the figure was given by M. Dandelin of Liege, in vol. ii. 

 of Quetelet's Correspondance Mathematique et Physique ; but 

 it offers nothing novel in principle to call for special remark. 

 Several neat solutions may also be seen in the Gentleman's 

 Diary for 1817; but they also differ more in form than in prin- 

 ciple, from others which had been previously published. 



XLIX. On a Difficulty in the Theory of the Attraction of 

 Spheroids. By J. MacCullagh, F.T.C.D* 



\ N approximate theorem discovered by Laplace, and re- 

 ■**■ lating to the attraction of a solid slightly differing from a 

 sphere on a point placed at its surface, has given rise to many 

 disputes. It has engaged the attention of Lagrange, Ivory, 



* Communicated by the Author. 



