150Author(s): William HooverSource: The American Mathematical Monthly, Vol. 8, No. 5 (May, 1901), p. 119Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2968623 .

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119

at infinity, the third vertex is the focus of the parabola into which the given conic becomes by projection.

The projection of the second triangle is a circumscribed triangle to the parabola, the circle circumscribing which triangle passing through the focus of the parabola, and proving the theorem.

The reciprocal theorern is: Two triangles are inscribed in a conic; their six sides touch another conic.

150. Proppsed by WILLIAM HOOVER, A.M., Ph.D., Professor of Mathematics and Astronomy, Ohio Uni- versity, Athens, Ohio.

Find the equation to a sphere cutting orthogonally four given spheres.

Solution by the PROPOSER.

Let X2+y2+z2 +2Ax+By+Cz+D-O.. ..(l), be the sphere cutting orthogonally the spheres

X2 +y +Z2 +2alx+2bly+2c,z+d,-O .... (2),

>Z2 + 2a x+d2 0 ...(3),

x2? +2a3X 3+d3 O .... (4),

x2? +2a4x + d4 0 .... (5),

Now, two spheres cut each other orthogonally if their radii and the dis- tance between their centers form a right triangle ; this requires, for (1) and (2),

(A-a1 )2+(B-b 1)2 (-c1 )2 -A B2?0+C2-D+a 12 +b12 +c12 -dl, or, 2Aa1? +2Bb1 +2Ccj-D-d1 0....(6).

Similarly for the intersection of (1) and each of (3), (4) and (5),

2Aa2+2Bb2+2Cc2-D -d:O=.. ..(7),

2Aa8+2Bb.? +2Cc3-D-d3O ... . (8),

2Aa4 + 2Bb4+2Cc4-D-d40 (9),

(6), (7), (8). and (9) give

2b1, 2cj, -1, d1

A A 2b2, 2c2, - 1, d2 (10), 2b8, 2c8, - 1, d3 2b4, 2c4, -1, d4

and like values for B, 0, and D, which in (1) gives the required equation. Similar demonstrations were received from J. W. YOUNG, G. B. M. ZERR, and LON C. WALKER.

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