Ailr action expressed iy (in algehdtc Quantity. 1(73 



Definition. 



Let (A) and (B) be portions, intercepted within tbe 

 hemisphere (H), of different cyhnders ; if the attraction of 

 (A) be equal to the attraction of (H) — (B), I call these 

 cylinders reciprocal as to attraction. 



Problem. 



To find any numler of reciprocal cylinders such that the 

 attraction of (A), or its equal (H) — (B), shall he an ulge^ 

 hraic expression. 



This will be effected, if the curves bounding the bases 

 of (A) and (B) be of such a nature, that the radii vec- 

 toresj drawn from the attracted point, are, 



for (A), 7-=2Rcos. 9{l — (1— cos.^"fl)^} 



for (B), /=2Rcos.6{l-cos. fl), 



and n be taken any odd whole positive number : for, by what 

 has been shown, in this and the former paper, we have 

 Attraction of (A) = Attraction of (H) — (B) = 



•— R/' cos. ''"'*"'$ . 3 the fluent to be taken from 9 = 0, to 



We may find cylinders whose portions, intercepted by 

 the hemisphere, have algebraic expressions for their at- 

 traction, by making r = 2R cos. 9 (1 — m cos. "9), or 



r = sR COS. 9 (1 —m sin. "" 6), and determining m in such 

 a manner as to eliminate the arcs from the expression of 

 the attraction. But this will not be so neat as the former 

 method, because there will be radicals employed. I shall 

 however give an example : 



Substitute, in (8), the first of the above-mentioned values, 

 and there arises 



F= 4 Rye + \ Rfco^. 29 . 9- 1 7«^- Rycos.^"""-!) . 9; 



herc 3ra-j-2 must be a whole even positive number, greater 

 than 2 : as to the integral, it must evidently be taken from 



«uch a value of 9, as gives cos. " 9 = — , to 9 = — . For 



' ° ?;t '2 



a particular example, let 3^+2 = 4, or n-= , ; then, 



* Tho^e cylinders will also be reciprocal, the equations of whose base* 

 •rt r=2R coi./'l 1 - a— jtn.^" ty \ , /=^2R cos. f (1— sin. '" /). 



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