Of GREJTES? JTTRJCTION. 217 



the height of that column, or of the cylinder, AD — x, AB, 

 the radius of the bafe, = r. 



By the lafl proportion, the column ftanding on DF, exerts 



on A an attraction in the direction AD, which is zz — ~XD 



X fini>. 



Now Ddzz x, Dfzz x z, and D d X D/rz xzx. Therefore 



• • • X X Z ' ' 



the attraction in the direction AD is — — x fin v zz x zfmv, 



x 



and reduced to the direction AB, it is x z fin v X cof z. 



This is the element of the attraction of the cylindric fhell 

 or ring, of which the radius is AD or x, and the thicknefs x j and 

 therefore integrated on the fuppofition that z only is variable, 



and x and v conflant, it gives * fin v j z cof z zzxfmv Xfmz 



for the attraction of the fhell. When z =r 90, and fin z — i, 

 we have the attraction of a quadrant of the fhell == x fin v, and 

 therefore that of the whole femicircle z= 2 * fin v. 



Next, if * be made variable, and confequently v, we have 



2 J x fin v for the attraction of the femi-cylinder. 



Now the angle v would have a for its fine if the radius were i/a 2 -{-x\ 



and fo fin v = , a , ■ ; wherefore the above expreffion is 

 Si a -f- x r 



/2 ax 



w , , a = 2 a L (x+va 2 -j- x 2 ) + C -, and as this muft va- 

 V a -{- x 



nifh when x = o ; 2 a L a -j- C = a, and C = — 2 L # , fo that 



the 



