Skip to main content

Full text of "157"

See other formats


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 

Read more about Early Journal Content at 
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 


157. Proposed by 6. B. M. ZEEE, A. M., Fb. D., Professor of Chemistry and Physics, The Temple College, 
Philadelphia, Pa. 

Solve the equations 
-?— +— y —+^—+— ?L_=i x i y i g i w =1 

(1+1 &+* c+r d+X ' a+n^ b+p-^ c+fi d+fi ' 

a + i' 6+y c-\-v d-\-v ' a-\-p b+p c+p d-\-p 

I. Solution by L. E, DICKSON, A. M.. Ph. D., Assistant Professor of Mathematics, University of Chicago, 
Chicago, 111. 

In view of the given equations, the following equation in t, 

a+t ' b+t ' c+t d+t 

has the roots X, /*, v, p. Setting f(t)~(t+a)(t+ V) (tf+c)(tf+ri!), we have 

*/«(0 + ^(0+*/o(0+«/<*(0=/(0, 
where f a (t) denotes the partial derivative of /(<) with respect to a. Hence 


is an identity in t. To obtain the value of x, set t=— a. Then 
-xfai-a^i-iyia+XXa+^ia + rXa+p), 
( a+ A)(a+ A )(a+T')(a + / >) 



The value of any other variable, say y, may be obtained similarly, or by 
interchanging a with b in the expression for x, as is evident from the symmetry 

This method of solution is equally simple for n such equations in n variables. 

II. Solution by L. C. WALKEE, A. II., Graduate Student, Leland Stanford University, Cal. 

By Art. 586 of Hall and Knight' s Higher Algebra, consider the following 
equation in 0, 

a + + b+d^~ c+6 d+0 O+0)(H 0)(c+0)(<H-0) ' 

x, y, z, u being for the present regarded as known quantities. 

This equation when cleared of fractions is of the third degree in 0, and is 


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+/.) 

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. 


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.