74 Mr. Atwood's Propositions determining the Positions 
v/ 1 — n ; and GH — ay ^== — V 1 — n : let P represent 
the centre of gravity of the area AEB ; then by the properties 
of the centre of gravity the following equation arises : 
GH x area EDCF = area ABDCFA x OH — area AEB x HP, 
that is 
a 3 y — s/ 1 — n = a ny OH — a 3 y JLziL 1 ; and conse- 
V 2 3 
fluently HO = l * 3 - VT 1 x ^ « x ✓ T x ~ ^ m which 
n J ViZyn 
quantity taking away the line HG = a x — s/ 1 — n = 
- ” there will remain the line GO = 
V18 X H 
X 3 — 3» — ^ IS x ^ 2 X 
A/18 x « 
Referring to the general expression, namely — 
AB s 
zD 
ds, we 
obtain in the present case AB = Sa 3 y 1 — ?i% D = a z n, GO 
^ a X 3 — 3» — a/i8x 1 — n* + a/ 2 x 
A/18 x w 
x 3 — 3» — a/T8 x + a/T x T^«|» . w } 1 j c j 1 
; wherefore — 
’ 12D 
d = 
8 a 3 X 1 — n 2. 
A/18 
quantity being put equal to o, in order to obtain the limit, 
and the whole being multiplied by 2 will give 2 v^Fx 
s / 1 — 72 — 3 — 3 </ 2 x V 1 — n -f 2 x v/ 1 — w, or 
a/ 1 — n — •— = ; wherefore 1 — n = fi or 71 — > ^ ie li- 
mit required. 
By the preceding determinations of the four limiting values of 
