104 Of such Portions of a Sphere as have their 



it contains some sulphates; and probably the sulphate c?f 

 lime, as the latter basis was indicated by the oxalic acid, and 

 the former by the sulphuric acid and the muriate of barytes. 

 I will not enter upon anv theoretical disquisitions ; but I 

 cannot help observing, that the presence of brimstone in 

 substances which not only can but actually do produce 

 hydrogen gas in such abimdance, has suggested to my 

 mind that sulphur itself mav be a product of thein, and 

 possibly only a modification of hydrogen. 



XVI. Of such Portiojis of a Sphere as have their Attraction 

 expressed by an algebraic Quantity. 



(Concluded from vol. xl. p. 329.) 



Sir, X ASSIGNED, in a former letter, such cylindric por- 

 tions as when taken from a sphere, or hemisphere, will 

 leave a remainder having an algebraic quantity for the 

 measure of its attraction. There is yet another problem ; 

 viz. to find the nature of the curve bounding the base, 

 when the attraction of the cylindric portion itself is an 

 algebraic quantity. It is scarcely necessary to observe that 

 equation ((S), which supposes the fluent, with respect to fl, 



to be taken from 9 = 0, to 5= -|^, is only adapted to parti- 

 cular cases. Let us take the general form : 



y^ 4^C(gR^03^j_( mcos..-.). ) (.^^^^^^ 



V C(2Rcos.<i)i (>Rcos. #)?) ' 



Or, 



3 •/ _ 3 y 3/ (ORcoS.^)! ^ ' 



A simple inspection of these forms will point out many 

 ways of effecting what we propose. 



Make, in equation (y), r = 2Rcos.ej 1 — (I— cos."fl)f? 



and it becomes 



^ 4 /.((2Rcos.%' (2Rco5. ^)l , ^. 1 _ 



F= -^fy 1-- '- (1-003."$)^ Cos. 6.6 = 



3" ^y^"^- "^ fl. 6j the integral to be taken from 6=0, to 



^ = Y' '' ^^ evident, that this will be an algebraic quan- 

 tity, as the problem requires, when n is any odd whole po- 

 sitive number. The curve, moreover, which bounds the 

 base of the cylinder is always algebraic. 



Defi' 



