V 
THE CATTLE-PROBLEM 
97 
ooloid cut off by 
lity in the cases 
the surface, and 
ios respectively. 
' —f p) 2 /h 2 is the 
vertical, 
jh the axis AM. 
s a point on CA 
along CA such 
hat MR = | CO. 
to AM from K 
BA 2 B 2 , BA 3 M 
1 to AM) as axes 
parabola BA.,B, 
above relation 
e Quadrature of 
Din 0 meets the 
d straight lines 
the other para- 
T and Q t U are 
le axis MA pro- 
\p) 2 \h 2 , there is 
(ii) if s< AR 2 :AM' 2 but >Q 1 Q Z 2 : AM 2 , the solid will not rest 
with its base touching the surface of the fluid in one point 
only, but in a position with the base entirely out of the fluid 
and the axis making with the surface an angle greater 
than U ; 
(iiia) if s — Q t Q 2 : AM 2 , there is stable equilibrium with one 
point of the base touching the surface and AM inclined to it 
at an angle equal to U; 
(iiib) if s = P 1 P 2 : AM 2 , there is stable equilibrium with one 
point of the base touching the surface and with AM inclined 
to it at an angle equal to T; 
(iv) if s > I\ P 2 : AM 2 but < Q x Q 2 ; AM 2 , there will be stable 
equilibrium in a position in which the base is more submerged ; 
(v) if s<P 1 P. 2 : AM 2 , there will be stable equilibrium with 
the base entirely out of the fluid and with the axis AM 
inclined to the surface at an angle less than T. 
It remains to mention the traditions regarding other in 
vestigations by Archimedes which have reached us in Greek 
or through the Arabic. 
• 
(a) The Cattle-Problem. 
This is a difficult problem in indeterminate analysis. It is 
required to find the number of bulls and cows of each of four 
colours, or to find 8 unknown quantities. The first part of 
the problem connects the unknowns by seven simple equations ; 
and the second part adds two more conditions to which the 
unknowns must be subject. If W, iv be the numbers of white 
bulls and cows respectively and (X, x), (Y, y), {Z, z) represent 
the numbers of the other three colours, we have first the 
following equations: 
(I) 
(H) 
1623.2 
W=ft + i)X + 7, 
(a) 
X = £ + ±)Z+Y, 
(0) 
Z =^ + i)W+Y, 
(y) 
W = (-| + (X + x), 
(5) 
x — (i + i) (Z + z), 
(«) 
2 = (HI)(F+2/)> 
(0 
y = {i + ^){W+w). 
(v) 
H 
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