328 On such Portions of a Sphere, &c. 



sider that portion which remains after penetration by the 

 cylinder, it becomes 



4 /-fiRcos. 9— r)f . ^ , , • • , 



F = — /- r- COS' S- ^ »• and this is to be an 



•iJ (2Rcos.9)i ' 



algebraic quantity. 



\. Suppose the curve ABFH, fig. 6. (Plate VIII.) to be 

 of such a nature, that the radius vector Am, or r = SR 

 COS. 9(1— sin. '"9) ; we get by substitution, 



t / (2Rcos. 9. sin.^"9)?- , ,• ^nf zA 



p _ _ /J — L COS. QJ— T -R / COS. * 9. 



3.7 (2R COS. 9)5 ^ ^ 



sin.s™ ^J .^1 R A/ sin. ^" 9. 9 - sin. =^" + -9. 9 J ; which 



expression will always be an algebraic quantity, when n is 

 an odd positive whole number. If we choose to express 

 the equation of this class of curves by rectangalajr co- 

 ordiiwtes, make An = x; mn^y; then Am = r= A/x' + y^'> 

 and putting these values in the equation •/•=2Rcos. J 

 (1— sin.~"9), we shall get 



In the simplest case, when ?z=!,this becomes (x*+y*)* 

 c= eRx' ; a line of the fourth order. 



2. Let r ss 2R cos. 9(1 — cos. "' 9), in which case 



F= ±.fJ2!^-2!l}l 00.. U=^Rrco..^'*H.L 



37 (2Kco...9ji 2 J 



This, like the former, will be an algebraic expression 

 when 71 is an odd p Jsitire whole number. The base of the 

 cvlindervvill liere have two p.'arls in the manner represented 

 by fig. 7. (Plate VIII.) The equation of these curves may 

 be expressed by rectangular coortbnates as in the last case ; 

 but having a few words to say on another subject, I shall 

 for the present quit this. 



X.Y. 

 P. S. — In rny letter of last month, respecting Mr. Le- 

 gendre's principle, I agreed with (hat mathematician, that 

 there can be no equation between c and A, B, C ; but then 

 I nmst be understood to mean, between those quantities 

 alone i " sans autre ancrle ni ligne quelconque ;" whicri i$ 

 all he professes to prove, and what everv clnld knew btt'ore. 

 But, who Lvcr drvanit that m a case of this kind it is ne- 

 cessary there should be an equalion between the variable 



quantities 



J 



