Skip to main content

Full text of "185"

See other formats


STOP 



Early Journal Content on JSTOR, Free to Anyone in the World 

This article is one of nearly 500,000 scholarly works digitized and made freely available to everyone in 
the world by JSTOR. 

Known as the Early Journal Content, this set of works include research articles, news, letters, and other 
writings published in more than 200 of the oldest leading academic journals. The works date from the 
mid-seventeenth to the early twentieth centuries. 

We encourage people to read and share the Early Journal Content openly and to tell others that this 
resource exists. People may post this content online or redistribute in any way for non-commercial 
purposes. 

Read more about Early Journal Content at http://about.jstor.org/participate-jstor/individuals/early- 
journal-content . 



JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people 
discover, use, and build upon a wide range of content through a powerful research and teaching 
platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not-for-profit 
organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please 
contact support@jstor.org. 



222 

satisfied by the four values $=X, $=/*, d=v, 8=p, in virtue of the given equa- 
tions ; hence it must be an identity. 

To find the value of x, multiply up by a + 6, and then put a +0=0; thus 

x= (a +A)(o+ Ji .)(a+y)(o+/.) 
(a—b')(a—c)(a—d) 

By symmetry, we have 

v= <fi+W+t>-)<b + v)(b+p) 
y (b-c)(b-d)(b-d) ' 

= _ (<>-M)(<>+/')(<>+*0(g+/») 
(c-d)(c-aXc-b) ' 

and U ~ (d-a)(d-b)(d-c) ' 

Similarly solved by Q. B. M. ZERR. 



GEOMETRY. 

186. Proposed by W. J. GBEENSTBEET, M. A., Editor of The Mathematical Gazette, Stroud. England. 

Given the tangential equations to two conies S, S' , find the tangential co-ordinates 
of the join of the poles of two given parallel lines with respect to S. Deduce the tangent- 
ial equation of the center of S, and find that of the intersection of S and S' . 

Solution by G. B. M. ZEEE, A.M., Ph.D., Professor of Chemistry and Physics, The Temple College, Philadel- 
phia, Pa. 

Let bc-f*=A, ca-g*=B, ab-h*=C, gh-af=F, Jif-bg^G, fg-ch=R. 

Then S=AX* +.B/* 2 + <M +2F/j.v+2GvX+2HX/i. 

Similarly, 8 , =A'X*+B' l i* + C'v* + 2F»v+2G'vX+2H'Xv. 

Let Aa+./i/3+vy and Xa-\-iJ.p+vy-\-m be the two given parallel lines; p, q, t 
and p', q', t' their poles with respect to S. Then for the first line 

ap + hq + gt=X, hp + bq +ft=v, gp +fq + ct=v. 

Solving these equations for p, q, t, 

p=(AX+Hfi+Gv)/ A , q=(HX+B,x+Fv)/A, 
t=(GX+Fv+Cv)/A, where A--=abc+2fgh-af*-bg*-cJi 2 . 

For the second line, 

ap'+hq'+gf+m=X, hp'+bq'+ft'+m=p., gp'+fq' + d'+m=v. 



223 

.-. p'=lAX+H r x+Gv-m(A+E+Q)y a . 
q'=[m+Bfi+Fv-m(B:+B+F)y a . 
t'=^lGl+Fn+Cv-m(G+ F+C)]/ a . 

(P> Q> 0> (.P't Q'> Q are the tangential co-ordinates of the join of the poles. 

Let A', B', C be the angles of the triangle of reference. The center is the 
pole of the line at infinity asinA'+/?sinB'-t-j'sinO'=0. The tangential co-ordin- 
ates of the center are obtained by substituting sinA', sinJB', sinO' for A, p., v in 
p, q, t and are 

8, =(As,inA'+HsinB'+ GsmC')/A , 

8 2 =(SsmA'+BsmB'+FsmG')/ A , 

8 3 =( <?sinA' + FsinB' + CsinC")/ a . 

.-. The tangential equation of the center is XS 1 -\-p8 i +y8 i =0. 

Write a + M' for a, b-t lib' for 6, c-\-kc' for c,f+lcf for/, g-\-bg' for g, A-f- 
hh' for h in AX*+BpL* + Cr*+2Ffiv + 2GrX + 2mft=0. 

Then the tangential equation of the four points of intersection of 8 and 8" is 
S+lc$-\-lc s S'=0 where 1c is undetermined, and 

$=(bc'+b'c-2ff)W+(ca'+c'a-2gg') ! x i +(ab'+a'b-2M')r* 

+2(gh'+g'h-af-a'flav+2(hf+h'f-bg'-b'g)7tl 

+ 2(fg'+fg-ch'-e'h)X,i. 

The condition for equal roots for Zr is $ 2 =4##', which is the equation of 
the four points of intersection. 

186. Proposed by J. E. HOT; Professor of Mathematics, Coronal Institute. San Marcos, Texas. 

If. two sides of a triangle and its in-circle be given in position, tne envelope of its 
circumcircle is a circle (Mannheim). [From Casey's Sequel to Euclid.] 

Solution by 6. B. M. ZEEE, A. M, Ph. D., Professor of Chemistry and Physics in The Temple College, Phila- 
delphia, Pa. 

Let vertex A be origin, sides b, c the axes. Then x i -\-2xyoosA+y s — bx— 
cy=0 is the equation to the circumcircle. Let this equation be written 

D-bx-cy=0....(l). 

Since the sides b, c and the inscribed circle are fixed in position, the tan- 
gents from A to the in-circle are constant. 
.•. b-\-c— a=& constant=m....(2). 
a=l/(6 8 -(-c 8 — 26ccosJ.). This in (2) gives after reduction, 

m 8 +2fc(l+cosA)-2m(6+c)=:0....(3).