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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. 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). 224 c from (1) in (3) gives 2b' ! x(l+eosA)+2l[my-mx-D(l+QOsA')~\+2Dm-m 2 y=0. The condition for equal roots of b is 2ic(l+cos J 4)(2Dm— m 2 y)=\my— mx— D(l+cosJL)] s or [Z>(l + cosA)— m(# + y)] 2 =2»» 2 a;#(l— cosA)=4m s ;n/sin 2 £.A. .-. D(l+cosA)— m(x-\-y) ±2m,}/ (xy)$,m^A=Q. .-. x*+2xy<nsA+y*-2^^-j[x+y±2(xy)smbA-\=0. .-. x 2 +2xy+y* -4xysm^A-2^rj[x+y±2 ] /(xy)siniA']=0. .-. lx±2^/(xy)smiA+y][xT2 l /(xy)sinlA -+y - 2eQg2 ^ ]=0- .-. xT2 v / (xy)siniA + y-^^j=0, or a; 8 + 2xyaosA-\-y 2 -\-\_m — 4m(a: + #)cos 2 £.A"|/4cos 4 £.l=0. This is the circle. CALCULUS. 144. Proposed by 6. B. M. ZERE. A. M„ Ph. D„ Professor of Chemistry and Physics, The Temple College, Philadelphia, Pa. Find the volume of the sphere, x % + y 2 + z 2 =2az, (a) within the paraboloid z=Ax 2 +By 2 ; (6) within the cone z 2 =Ax 2 +By 2 . Solution by the PROPOSER. a; 2 +# 2 +z 8 =2az....(l), z=Ax 2 +By 2 ....(2), z 2 =Ax 2 +By i ....(Z). From (1), z=a± \/ {a 1 -x s -y 2 }=a ± y {a 2 — r 2 }. From v= I I zrdrdO we get v^iC" f y (a* -r 2 )d$rdr=% f V'-Ca 8 -*")* ]d0. J o J «^ (a). From (1) and (2), x 2 +y 2 + (Ax 2 +By* ) s =2a( Ax 2 +By 2 ) . r s +r 4 (Acos s + Bsin 8 0) 8 =2ar s Gicos 2 0+£sin 2 0).