as are terminated by Planes , &c, #79 
4>dL . — +k ; and the fluent, taken from x = o to x = k, is 
A = idh 
which agrees with what Mr. Playfair found in a different 
manner ; except in this case, the object of the present propo- 
sition is different from that of his, which finds the action of 
such portions of the cylinder as have sectors for their bases. 
Prop. 29. 
Let the base of the cylinder be the figure parmnbp, fig. 17, 
the curves par, pbn being inverted parabolas ; or in which 
pm 1 = 2 x rm. Let the attracted point be at p ; and let the 
cylinder be extended to the distance d above and below the 
plane of the figure. 
Using the same formula as before, and putting pm == x, we 
have b = d, b' = — ; and, for the action of the solid, 
7 ot , * 
dx \ 
- ■ ■■ - ■■■''— or 
od’d 2 -j- a, z x z -f- X* • 
“ 4/x “ c (tang - = 
the last term of which becomes, by taking the fluxion of the 
arc. 
A = 4 fx arc (tang. 
A == 4a: arc (tang. 
dx 
-4 fj 
a. z d 3 xx — dx ! x 
| a. z d z -\-(ct, z -\-d z )x z -frx* ^ & z d 2 <z z x z -t x* 
an expression integrable by circular arcs and logarithms* 
When d is infinite, this fluent takes a very simple form, viz. 
2XX 
~ = - s*L (1 + and > in 
00 
O o a 
