73 Mr. Atwood’s Propositions determining the Positions 
and the specific gravity = n when that of the fluid is equal to 
1, also make VW or XQ = a ; then QR == at , and On — ~ 
X?z = \ f a. -f- — , or Xw = — x v ^ 4 “1“ 
To find the sine of the angle wXR, make the following 
proportion. As 'Rn of Qri (^) : Xrc (-7 x v/ 4 -f f ) : : sine 
nXR : sine XR« : wherefore sine ?zXR = — . e - ” RX -^-L < 0 r be- 
‘S 4 + f* 
cause sine wRX = -- 1 ■ , sine «XR = — — ■ ■ — — — — . ; 
v'i + V - *"4 + V x v'i + v - 
cos. wXR = + 1 ■ — : and since X^ = — x Xw = 
^4 + ^ X V'i + 3 
+ it follows that Xt = +«•*; + «• _ i x 
3 3« ^4 4 z 1 x V^i + i 2 3 
; and since the triangles XPZ, XOR, as also the tri- 
angles ZXm, RXw, are similar and equal, the line X6 = Xc ; 
and consequently be = 2 X<; 
za x 2 -4 t 1 
3 x v'T+T 
; which quantity = 
6 in the general value — <£?. And since the specific gra- 
vity of the solid is = n, the height SL = c, and the base CD 
= 2 a, the immersed part or ACDB = 2 acn, which in the ge- 
neral expression is denoted by V ; and the volume QXR =a 
is denoted by the letter A in such general value. 
Substituting, therefore, in the expression — ds, 
for b ; for A ; and 2 acn for V ; the distance between the 
vertical lines passing through the centres of gravity of the 
solid and the centre of gravity of the part immersed, appears 
