106 Of such Portions of a Sphere as have their 



— 7»=sR/cos. i9 J =s -^/s cos.^fij = -^/|36 + 



4 COS. 26 . fl + COS. 49 . 9 L and F= (^^- — Wi^) Ry 'S + 



(1 _rnV Y R/cos. 29. 9- ^ycos. 49. 9. 



4 



Here the arcs are avoided by making m^= —, or m = 

 /i-^j whence, F= - ^ Ry| cos. 29.9 + cos.49.fl| 

 = — — R sin. 29 — ^ R sin. 49. 



9 9 



This is the attraction of the portion of such a cylinder 

 as has, for the radius vector of its base, r=2R cos. 9(1 — 



(-^^ cos. ^9). The base of the cylinder will plainly 

 consist of two parts like fig. 3 of the former paper. 



The equation (—)^ cos. ■3"9=1 gives cos. 6= -^ ; so 



that each portion of the base lies between 30 and 90 de- 

 grees, on each side of the diameter, passing through the 

 attracted point ; and, within these limits, 



F=-Rx-^ + -R X --2-- ^R— 2--^^. 



Having terminated what I meant to say respecting the 

 nttraction of this kind of solids, I will add a word or two 

 concerning their solidity. 



Let fig. 4 (Plate III) represent the base of a hemisphere, 

 A its centre, ABCG a curve, whose parts, on each side of 

 the radius AC, are equal and similar. Put R -AC, 

 r =AD, 9 = the angle DAC. If we conceive a cylinder 

 ■trected on the base ABCG, the solidity (S) of the part 

 intercepted within the hemisphere is evidently S = 



Qj/v R^—r"^ I rrd', or, taking the fluent, with respect to 



Tj so that it may vanish when r=0, 



S= |R^9-|/(R^-r0^9 (0. 



Make, in (g), r^ = R* (1— sin." 9) ; and, because the 

 curve denoted by this equation is contained in half the base 

 of the hemisphere, it becomes 



S=.tRB^AR3ysin.'"9 9. 



Kow, the first terra -^ R* is the solidity of one half 



of 



