and Strength of Materials. 283 
Retaining the last figure, fig. Sth, and the 
same notation, and supposing the side ABC 
to be that of tension, we have the arc ACB 
= A; and if G and P be the centres of gra- 
vity and percussion of that arc, and P’ the 
centre of percussion of the are of compression 
BDA, the formula gfe, will become 
sX (EG) x Ax (EP-+EP’), But OG= OCx AB 
aL.C are ACB | 
(Dealtry’s Fluxions, Art. 64, Ex. 12))are 
Orb : 
and since OK = r—a, «.. EG = a — (r—a) 
Serb —(r—a)A. 
Tae of an ae 
And assuming EP+EP’, or PP’, = twice 
the distance of the centre of percussion of the 
semicirculararc IDH from O, (for the same 
reason as in the last example), and calling 
that distance = P, any distance OR =a, the 
arc DS = z, and supposing the arc ACBDA 
indefinitely thin, we shall have 
7 up — rary 
v2 a ns 
P= — @'—2*)" since 2 LS pcliy 
S®: are (7?—a:*) 
(r?—a?)* 
(Dealtry 44). But the correct fluent of 
— rary 
(r?—a?)t 
='785398 &c. X r3. Inlike manner, f — ree 
(r?=—?)2 
(when z= a quadrant, or 2 = 0) 
(when x=0)=r*. And hence 
