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166Author(s): Marcus BakerSource: The American Mathematical Monthly, Vol. 10, No. 1 (Jan., 1903), pp. 14-15Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2971135 .
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14
Hence there are six sets of solutions:
X-0,j y-0; X, - W3 p - 02 m m
(2) Solution of the sinmultaneous equations
mx4 +py=O, my3 +px3 =0.
In addition to the solutions x=y=O, there are exactly nine sets of solutions
xiphr9Mnt-4)ho f yu-n4p7/9M(-7)/t,
wvhere e is an arbitrary niinth root of ninity.
165. Proposed byJ. K. ELLWOOD, A. M., Principal of Colfax School, Pittsburg, Pa.
Solve X4 -x= 14, by quadratics.
Solution by G. B. M. ZERR, A.M., Ph.D., Professor of Cbemistry and Physics, The Temple College. Pbiladel- phia, Pa.
X4-x=14 or $'-16=x--2.
___x _ x 2
2 _- _ 2 1 . 4 X2- +s 4 + 2+
x8 +4+ 4(x2 +4)2 = 4 2+4? 4(xe + 4)
1 x--2 or x=-2 + 4
x-=2 or x? + 2x2 +4x+7=0.
.x2 or---1l nearly, or - 1 ?1?//(-843)] nearly.
166. Proposed by MARCUS BAKER. U. S. Geological Survey, Washington, D. C.
Solve ax + by -zx ....(1). cy+dz=-2xy .... (2). ez+fx-2yz .... (3).
Solution by the PROPOSER,
From (1), (2), and (3), respectively,
(2z-a)x dz ez +fx Y b 2x-c 2z
whence x(2x-c)(2z-a)=zbdz .... (4), fx(2x-c) +ez(2x-c) =2dz? .... (5).
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From (4), z=2xa (2x ) bd which substituted in (5), gives, after reduction,
x' + 1 (ae-2cf)x8 + [fc?-a2d d-2(ace + bdf )]xi
2f 4f
-1 8j [ac(ad+ce) +bd(2cf-ae)]x +tj!bd(ace + bdf ) -O.
Similarly,
Y + 21 (ac-2be)y3 + -[f Uc?-e2b-2(ace+ bdf )]y2
1 ~~~1 + 8b [ce(ae+cf ) + df(2be-ac)]y+16 lb df(ace+bdf )=0.
Z+2d (ce-2ad)z3+4d [a2d-e2b-2(ace+ibdf)]z2
1 1 + 81 [ac(ae+be) + bf(2ad-ce)]z +16d bf(ace +Ibdf ) O.
Also solved by LON C. WALKER.
GEOMETRY.
REMARXS ON No. 187, GEOMETRY, BY J. R. IIITT, Goss, Miss.
There seems to be an error in (4) of Professor Zerr's demonstration of
No. 187, Geometry. The result given is not correct. For t== 1 bsinCoosC, t
= 1 -boosC, t -7 = bsinC, from which it is seen that in general P cannot equal I/ 2bcoC,t /2
t,t2. It is also easilyseen that if t: t _t: tt, then must DI=b, whereas DI cannot be >kb.
CALCULUS.
154. Proposed by B. R. DOWNER, HopkinBville, Ky.
At the equinox, when the sun is on the celestial equator, a mat starts driving on a perfectly level plain at six o'clock in the morning, and continues, going always from the sun, at the uniform rate of six nmiles per hour, until six o'clock in the eveninig. Required the path he will travel and the distance in a str aight linie from starting point to stopping point.
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