MATHEMATICAL SECTION. 149 
Er1egHtH MEETING. DrEcEMBER 5, 1883. 
The Chairman presided. 
Fourteen members and guests present. 
Mr. ALvorpD discussed 
A SPECIAL CASE IN MAXIMA AND MINIMA, 
the problem being to find the radius of the sphere that will displace 
the maximum quantity of liquid from a conical wine glass full of 
water. 
The differential co-efficient, when put equal to zero, is in the form 
of two factors. Equating each to zero, one gives the radius of the 
maximum sought; the other gives a still larger radius, which proves 
to be the radius of the sphere just tangent to the centre of the base 
of the cone, and to the sides of the cone, extended upwards. This 
gives the minimum displacement equal to zero. Calling a the 
radius of the base, 6 the height, and ¢ the slant height of the cone, 
theradius of the sphere producing maximum displacement equals 
abe 
Ce — aa fe)? 
ment equals 
the radius corresponding to minimum displace- 
C— a. 
When the radius is still greater, the sphere does not reach the 
surface of the liquid, but displaces an imaginary quantity of the 
same. An analytical expression for this case was sought in vain; 
the result above is simple, and no square root of a negative 
quantity appears. By some deyice in the mode of investigation, 
this imaginary case might appear, as in the question to obtain the 
radical axis of two circles, discussed by Salmon. 
Mr. KuMMELL suggested that the close relation between the 
circle x” + y’ = FR’ and the equilateral hyperbola 2? — y’?= R’, each 
of which could be regarded as an imaginary branch of the other, 
might help us to understand many of such difficulties. He showed 
that the radical axis of two circles not intersecting was the com- 
mon chord of two equilateral hyperbolas whose major axes were. 
those diameters of the circles which lie in the same straight line. 
Mr. EvLLiottr read a communication on 
A FINANCIAL PROBLEM, 
in which he gave formule for calculating the advantage of in- 
