326 Of such Portions of a Sphere as have their 



issue, no humiliation ought to be experienced, a5 the 

 parties will monk no leaiher. 



1 have the honour to be, dear sir. 

 Your most faithful servant, 



George Pearson. 

 P. S. — I have no where charged Dr. Marcet in the terms 

 alleged, viz. that he had comn)itted lluvders in reasoning. 

 I can well spare that word llimder from my vocabulary, 

 having little, use for it, although by the law of retaliation 

 amply justifiable. 



LV. Of svck Porfiom of a Sphere as have their Attraction 

 expressed hy an algebraic Quantity. 



JL HE measure of the surface of a sphere involvirig in its 

 expression the lengtli of the circumference of a ciiclc, it 

 has been thought an interesting problem, even so long ago 

 as the time of i^appus, to assign such portions of that f-.wr- 

 face as ad'uit of an exact quadrature: and th.s kiud of 

 inquiry hu'^ i)een extended by Euler, and some iater writers, 

 to the finding such parts of the solidity as may be exactly 

 cubed. 



Now as the attraction of ;i sphere, on a point at its sur- 

 face, is exprc>sed hy the same kind of transcendental quan- 

 tity as the surface, or the s<diduy, it seems a probiem 

 equiilv intercstuig with those above mentioned, to deter- 

 i)i;ne such portions as have their attraction an algebraic 

 quantity. 



' Let the circle AtFG, fig. .5. (Plate VIIT.) be the base 

 of a hemisphere, AF a diameter, ABCH a curve touching 

 or mettiiig the circle at A, and having its portions ABC, 

 AHC equal. and similar to one another. Conceive a risht 

 cylinder to be erected on the base ABCFI, and to pass 

 thrwui>h the surface of the hemisphere ; it is required to 

 find the attraction of that portion of the cylinder, that 

 is intercepted by the hemisphere, on a point at A, in 

 the direction AF. Let a line AD, drawn frorn A to 

 any point D In the base, be called r; the anoxic DAF, 9; 

 Let f (r, fi) represent a perpendicular to the base at D, 

 and extended to the surface of the hemisphere. Then, if 

 F denote the force of the cylindrical portion in question, 

 it is easy lo see that 



