83Author(s): William HooverSource: The American Mathematical Monthly, Vol. 5, No. 2 (Feb., 1898), p. 46Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2968566 .

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46

of R, {-[I(s-a) + j(s-c)cosB], - (s-c)sinB};

of Q, {i[s+a?(s-b)cosC], -i(s-b)sinC};

of P, { 0[s(cosC? cosB)-a], s(sinC+ sinB}.

BH=FrcotkB, OI=r, bE=x, BE=xcotlB.

.(r-x)2eot 2B B+ (r-X) 2=(r x)2 . *. x-=r(1-siniB)/(14 sinlB). Let r(1-sin?B)/(1 + sin4B)--r, r(1-siniC)/(1 + siniC)=n, r(1-siniA)

/(1 + sinjA )=1.

*. codrdinatesof bare (vicotlB, mn); of c, (a-ncotiC, n); of a, (pcosaBC,

psinaBC), where p=z/[c2-2clcotIB + 12cosec2IA], and tanaAB==(ctanB- ItanB

cotjA-1)/(c-1cot1A +1 tanB).

Substituting in (1) atnd (2) the trutlh of the proposition appears. By substituting the co6rdinates of P, Q, R, and a, b, c in (1), (2) we get,

after a prodigious amount of work, the co'ordinates of two points. If the line through these two points coincides with the line through 0, M, the proposition is true.

[NOTE.-Dr. Zerr furnished a very beautiful figure to go with his solution, but we lacked the time to engrave it. EDITOR.]

83. Proposed by WILLIAM HOOVER, A. M., Ph. D., Professor of Mathematics and Astronomy in Ohio Uni- versity, Athens, 0.

0 being variable, find the locus of a point whose coordinates are atan(O+tf a), btan(O +f/).

Solution by the PROPOSER.

The rectilinear co6rdinates being x and y, x=atan(0+ r) .......... (1), y=btan (0+13) ........ (2). (1) gives 0+a zztan-'(x/a) .............. (3),

0+ /3ztan(-1(/b) .. . (4). Eliminating 0, tani-'(x/a)-tan-(y/b)a-/o .... (5).

Taking tangents of both members of (5) and reducing,

xy-cot(c-3)(bx-ay)+ab-. .(6),

the equation to the required locus. Solved in a similar manner by COOPER D. SCHMITT, T. W. PALMER, OTTO CLAYTON, and G.

B. M. ZERR.

CALCULUS.

65. Proposed by GEORGE LILLEY, Ph. D., LL. D., Professor of Mathematics, State University, Eugene, Ore.

A string is wound spirally 100 times around a cone 100 feet high and 2 feet in diam- eter at the base. Through what distance will a duick swim in unwinding the string keep- ing it taut at all tinmes, the cone standing on its base and at right angles to the surface of the water ?

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