576 
PROBLEMS AND SOLUTIONS. 
[705 
[Vol. xyil, p. 60.] 
3677. (Proposed by Professor Cayley.)—Find at any point of a plane curve the 
angle between the normal and the line drawn from the point to the centre of the 
chord parallel and indefinitely near to the tangent at the point; and examine whether 
a like question applies to a point on a surface and the indicatrix section at such point. 
[Yol. xvii., p. 72.] 
3564. (Proposed by Professor Cayley.)—To determine the least circle enclosing 
three given points. 
[Vol. xviil, July to December, 1872, p. 68.] 
3875. (Proposed by Professor Cayley.)—Given the constant a and the variables 
x, y, to construct mechanically a - ■ X -; or what is the same thing, given the fixed 
points A, B, and the moving point P, to mechanically connect therewith a point P 
such that PP' shall be always at right angles to AB, and the point P' in the 
circle APB. 
[Vol. xx., July to December, 1873, pp. 106, 107.] 
3430. (Proposed by W. J. C. Miller.)—Find the equation of the first negative 
focal pedal of (1) an ellipsoid, and (2) an ellipse. 
Solution by Professor Cayley. 
1. It is easily seen that if a sphere be drawn, passing through the centre of 
the given quadric and touching it at any point (x\ y', z'), then the point (x, y, z) on 
the required surface, which corresponds to (x, y', z'), is the extremity of the diameter 
of this sphere which passes through the centre of the quadric. We thus easily find 
the expressions 
x = x '{ 2 ~a’)’ 2 = / ( 2 -?) ; 
where 
t = a/ 2 + y'~ + z'-. 
Solving these equations for x, y, z', and substituting in the two equations 
xx + yy' + zz = x' 2 + y- + z\ + | 7 + ^ = 1,