524 PHILOSOPHICAL TKANSACTIONS. [aNNO 1732. 



wards c be given, and = /;, and the centrifugal force at a also given, and =zf. 

 Put AC = a, cy = b, eg = r, and the sine of the angle dcp = /(, to the 

 radius = 1 ; then will gl = hr, and from y demitting the perpendicular yu on 

 the radius cd produced, it will be or = bh, and yG = (by Eucl. ii, 12) 

 ^{bb + lbkr-\- rr). 



Now since the gravity at a towards y is = tt ; by saying -rr : tt' :: {a + b)"' : 



(bb + 2bhr + rr) h'" , the gravity at g or tt' = — — ^— ^ " ^ . 



> . • 1 1 / /, X (bh + ib/ir + rr) 5" 



And to derive that towards c, say tt : tt :: cy : gr, or — ^ —■ — - — : tt 



:: {bb + 2bhr + rr)h: bh -\- r; hence is had the force from the attraction to- 



, A ■ A, 1 " 'T {hh + r) . (bb + mr + rr)''^'" - '^ 

 wards v, derived towards c, or tt = — --rr- . 



'' (a + 6)"' 



There will further be had, since the gravity at a towards c\s = p, the gra- 

 vity of g towards c = ^; therefore the whole gravity towards c, arising from 



, , .. , , .,, , -T (bh + r) . (bb + 'ibhr + rr) h'" - k pr" 



both the gravities towards y and c, will be — ^^ —^ „ h ^ . 



Now since the centrifugal force at a is = /; by saying/:/' :•. a -{■ b : b -\- hr, 

 there will be the centrifugal force at g, or/' = - ^ ^ ''^ ; and to find the part 



of that force which draws towards d, make/' :J" :: gh : gk, or ^ aXb '-^ '' 

 1 : A ; hence there is had the force opposite to the gravity towards c, or/" = 



fh (b + In) 

 a + b • 



Therefore the force towards c, resulting from all these forces, will be 



ff (be + r) . (bb + "i bhr + rr) > " i , P]2. . /^ (''' + ^r) 

 (a + b)'" "^ n" a + b ' 



Conceiving then, as in the first problem, the column CD, composed of an 

 infinity of small cylinders r, there will be 



/^(b/i + r).(bb + 2b/>r + rr)i'"-i , pr" Hi (b + /ir)\ • ,. , ,, ,, 



F (— — , ,,,„ — h ^ — — -7^, -) r, which must be equal to 



\ (a + b)"' 'a" a + b J T 



some constant weight. Therefore 



^(bb+ 2bhr + r/-)^'"-^ , pr" + ' _ /Mr _ fhkrr _ 



(m+ l).(a + 6)"' "''(«+l).a'" a+b 2(a + i)~'** 



To correct this equation, when A = 1, it must ber = o; then there will 

 result-^^^^^ — , -| ^^ — --^ , / " , , = A. And the equation corrected will 



?H + 1 ' »( + 1 a + b 'l(a + b) ^ 



, ^ (bb + 2bhr + rr) s'" + g p r'' + ' J'hhr _ fhhrr _ -(ci + b) pa_ _ fab _ fua 



°^ {m+\).(a+b)'" "*"(«+ 1)«" a + b 2(« + /.)~«(+l '^n + 1 a + b 'i(a + b)' 



Or, by writing c for a -f- b, and ^ for (m -f l) X (« -f 1); then 1 {n -f 1) 

 TTfl" {bb + Ibhr -f rr) ^^ + I + 2 {m + J ) pd'h- + ' — 2 (jfcC be'" - ' hr — qfa" C" - ' 

 hhrr =1{n-\- 1 ) ttg" C" + ' -f 2 (m -f 1 ) pa" + ' C" — 'Icjja" + ■ be'" - > — qfa" + ^ 



