102 Mr. Atwood's Propositions determining the Positions 
rate the cases in which the solid floats permanently with the 
base entirely extant above the fluid's surface, from those in 
which a part of the base is immersed under it. 
The notation remaining as before, since OB or BN 
\ a — 3 p — 4 aV ii 
and EB = a , (fig. 26.) by addition EN 
; and because NW = CQ* = a Vh, it 
follows that EW = EN — NW 
10 a — 3 p — 10 a V 11 
: and since 
AE = \/ ap, the tangent of the angle CNO or AWE, is to 
radius, as EA to EW, or as s/ ap \ 
— 10 a — 3 p — 10 a Vn 
, that is. 
making the radius = 1, the tangent of the angle AWE or CNO 
= ^ . but the tangentt of CNO = v/ 
10a — 3/>- loaVn ° 4 « — T>p-\aV n 
which two quantities are therefore equal, or — - - at> * 6 
10a — 3 p— 10 a V n 
™ ^ap X 6 
10 ma — 3 p 
v/— : 
or 
if 1 — s/ n is put = m, 
4 ! a — 3 p — 4 a Vn 
— ./ ? — • and by squaring both sides, r t ^ 1 — rr 
v ^ma — 3 /> ’ J ^ o ’ ioom*a x — 6oma p -f 9 p x 
— J ? — or a v H “- — - z = — - - , which is re- 
2 x 4^7- 3i>’ 1 oom a ~ oomap + 9 p \ma — 
duced to the equation, 100 wl a — 6om ap -J- yp 1 = 96m cf 
72 «f ; or — ^Toot- 9 " - * m = Wherefore m 
30 pa + 4 8aZ , . ^ / 3 ou P + 4 8ai T 
100 aP 100 aP I 
9 P Z + 7 __ SQp + 4 8 ^ 
100 aP 100 a. 
* By the preceding investigation in appears, that HO — — - , and since HO = 
GC and i GC, it follows that CQjor NW = n . 
f Page 101. 
