200 Wisconsin Academy of Sciences, Arts and Letters. 



itesimal, the point x of intersection of chord g d with C b coincides with 

 point a, and a b equals 1 n. 



Draw t a perpendicular to P C. Then angle C P a equals t a C. Let this 

 angle be represented by 5. Then is circumference or length of arc gener- 

 ated by point 1 rotated on axis P C for one rotation or an infinitesimal 

 part of a rotation to circumference or length of arc generated by point a so 

 rotated, as unity to cosine 5. 



Divide the equal chords g d and h k each into the same number of equal 

 and infinitesimal parts, and from the points of division draw lines as s e, f c, 

 etc., parallel to C a, and y o, z p, etc., parallel to C A. Then for one or an 

 infinitesimal part of a rotation, the volumes generated by surfaces having 

 dimensions a b, s e, f c, etc., each into an equal and infinitesimal division 

 of length along g d, are directly proportional to squares of distances P a, 

 Ps, Pf, etc. In case thenof a sphere of homogeneous density each division 

 along chord g d has a mass directly proportional to the square of its dis- 

 tance from P. Therefore per law of attraction for particles directly as 

 mass and inversely as square of distance,'th.e attraction of the mass at any 

 division along chord g d, on particle P is the same as that at division a. 

 The attraction then, in direction P a of whole mass between chords e c and 

 g d, is equal to that of mass at division a into the number of divisions of 

 chord g d into inverse square of distance P a. 



Let mass of sphere between chords op and h k rotated be represented by 

 m. Then because h k equals g d, and 1 n equals a b, and because circum- 

 ference of point 1 to circumference of point a rotated is as unity to 

 cosine 3, mas3 of division a into chord g d equals m cos 5. In right 

 angled triangle Pa C let hypotheneus P C be represented by D. Then side 

 Pa equals D cos. 5. 



Attraction of mass (m cos 3) on P in direction Pa= 

 Attraction of mass (m cos 3) on P in direction P C= 



m cos 3 

 D- co=^ 3 



m cos' js- m 



D' cos- 3 D'= 



But — equals the attraction of a mass (m) condensed to the size of a par- 

 ticle at the center of the sphere on particle P. 



Because P C is the axis ot rotation of choi'ds e c and g d, particle P must 

 be attracted in direction P C, and with a force equal to the attraction of 

 the mass cut from the sphere by one rotation of the chords o p and h k, 

 condensed at the center of the sphere. 



As n 1 is any part of the radius A C the whole of a homogeneous sphere 

 attracts an outside particle the same as the mass of the sphere con- 

 densed to its center. 



4. A spherical shell is a sphere less a sphere of smaller radius. In ac- 

 cordance then, with investigation of Art. 3, a spherical shell attracts an 

 outside particle the same as the mass of the shell condensed at its center. 



