Attraction expressed ly an algebraic Quantity. 107 

 of the hemisphere j consequently, the other part, viz. 

 •^ R' /sin. " 6 . 9 is what iemaius of the half hemisphere 

 after taking away the included portion of the cylinder, 

 whose base is defined by the equation r^ = R^ (1 — sin.""fl): 

 and this expression of the solidity will always be algebraic, 

 when n is an odd whole positive number. Let us now 

 find the equation of these curves in rectangular coordinates. 



Putx=Ara, y = mn; then r- = R- (1— sin. '^"d) becomes 

 a- + 2,^ = R^(l ^)j or, (x'-+ f-f^ = 



R* -j (x^ 4- 2^^) ~ y\ » where n is an odd positive whole 



number. When m =1, x' + y^= Bx, and the curve is a 

 circle : This is the ivell-known case of Bossut. 



Problem. 



It is required to assign the bases of cylinders, which 

 may be the reciprocals (as to solidity) of those alreadv found: 

 that is, whose portions, included within the half hemi- 

 sphere, may = -"- R' Ain. " fl . 9. 



This will be effected if we put, in equation (s), r^ = 

 R^ J 1 _(1 _ sin.^'9)t J , for there results S =: |- R' 9 - 



-|Ry(I- sin. ^"6) J = — R^ysin.^"9.6, the fluent to 



be taken from 6 = o to 9 = -^. If we want these cylinders 



to have their included portions algebraic, 7i must be aa 

 odd whole positive number. 

 I add another 



Problem. 



Assign the base of such a cylinder, as shall have an cdgfi' 

 Iraic expression fur the sulldity of that portion luhich is in- 

 tercepted in the half hemisphere; and shall sotisfi/ tlve 

 further condition, that this intercepted solidity shall ap- 

 proach as near us we phase to thai of the half hemispliere 

 itself. 



Make, in equation (e), ?' = R* | 1 — (I — cos.729)*|, it 



becomes 



S = |- R' 9 - -| R'y - cos. n&)&= I Rycos. nd.&^ 



VR» 

 3«" 



