323 



KNOWL-EDGE 



[Feu. 10, 1882. 



foand tlint it is jiroportionni to 2-16 of tho name ; whilo tho capacity 

 iif liinKB iH iircii><)rtioiinl to 2 -l of tlie lioiRlil for boys, nnd to tlic 

 Hiiuiiro of tlu> lu-i(flit for K'fl"- •'■'• NuKi)rBky'n ronoarelioH will noon 

 lir publiMhcd in ii Bcparati' work. .\8 to tlio r»'lntion between the 

 wrii^lit of nmii unci tlie capacity of lun^". it is tolerably iiormnnent, 

 iiud iui viiriutioiiH are inontly due to dillurenccB in llie amount oi 

 flit in tlie bodied of different men. 



K.IPLOSIO.N OK Aqi'a Ammoni i:.--Tlie I'hariiiareulical Joiir.tnl 

 iiionlH a recent ea«o of on expUwion of ordinary liquor ainmoniie, 

 followed by serious resnItH. A Ilelfast woman, subject toheadacli'', 

 Bent her daughter to the drii(;iti»l to purchase a small quantity of 

 "head salt.s," for which he guve her liquor animoniu', or " Hpirit of 

 hartshorn," inslead of tho wilt, carbonate of ammonia. The rial 

 was jiut on a shi'lf and not used for n few days. Ilavinif a head- 

 ache, tho woman lifted tho romody t > apply it, and had it inh- ' 

 hand for a few minutes only when the vial suddenly cxjiloded, 

 NcatterinK the contents over her face. Her eye was destroyed, and 

 her month and throat burned, the skin of both having been torn off. 

 Tho vial had been put on the mantelpiece previous to the time it 

 was used, and when about to apply the contents the woman was 

 sitting near the tiro. 



A NEW work, by Jlr. Richard Ifcado, Assistant Keeper of Mining 

 Uocords, entitled '"Tlio Coal and Iron Industries of the United 

 Kingdom," will be issued about the loth inst., by Messrs. Crosby, 

 Lockwood, & Co. Besides a description of the coal-fields and the 

 )irincipal scams of coal, Mr. Meade's book will include an account of 

 the occurrence of iron ores in veins and seams, and a history of the 

 rise and progress of pig-iron manufacture since tho year 1740. 

 .Maps illustrating the position of coal-lields and iron-stone deposits 

 throughout the kingdom will accompany the work. Messrs. Crosby, 

 l.ockwood, & Co., will also issue, during the month of February, a 

 new work, by Mr. Lowis D'A. Jackson, author of " Hydraulic 

 .Manual and Statistics," entitled '• Modern Metrology." This 

 manual will treat of the metrical units and systems of the present 

 century, and will include an ap()cndix containing a proposed English 

 system. The book will, we understand, be dedicated, by permission, 

 to the Risrht Uou. W. E. Gladstone. 



0\iv i*latl)fmatical Column. 



MATHEMATICAL QUERIES. 



[29]— An ellipse has semi-diameters DB, DF (DB = 2DF) . From D, 

 /)C is drawn, making an angle cot" '2 with DIS (on the same side 

 as DF), and DA making angle .4r»C = angIe CDB. Make DA = DB, 

 and draw ACB. From AD cut off 4E = one-fourth AD. With 

 centre E describe circle AGH, cutting elliptic quadrant AF in H and 

 (.' ((t nearer to F). Draw GI perp. to BD. Let A/ cut DC in J. 

 With centre B and radius CJ describe circle cutting CD in A', and 

 with centre C describe circular arc KL, cutting KB in L. It is 

 required to determine geometrically the ratio of the arc KL to the 

 [Straight line CB. — Edmv.n'd Hunt. 



[30] — Simultaneous Equations. — 



x' + y = 11 



y^ + X = 7. — Thomas Fawcett. 



[ This equation can readily be solved, so far as finding the obvious 

 roots is concerned ; on the other hand, if a and b be written for 11 

 and 7, the equation cannot be reduced to a quadratic. We leave 

 the equation in the above form, as an exercise for the young 

 reader.— Ed.] 



[31]— Equation. — 



a;' + 4x' = 27. — W. H. B. 



[32] — How TO Analyse a Cukve ? — A curved object has to be 

 reproduced in different sizes. A tracing of its curve has been made 

 on paper. How can an analysis and definition of the curve most 

 easily be arrived at ? A base line has been drawn on the convex 

 side of the cnr\'e and offsets taken to the curve as noted below : — 



Baseline -15 '25 -35 '6 -7 -85 



Offsets 2-3 1-7 1-5 13 1- '9 8 



Base Hue 10 12 1-15 1-75 1-95 21 



Oftaets 7 o -1 35 '35 



Baseline 2.1 2G5 30 3-5 41 



Offsets 2-25 2 15 1 05 



The measurements being in inches and decimals, can the law or 

 laws of the course of the curve be deduced from these figm-es ? If 

 yes, how ? — Fred. W. Foster. 



[33] — Bisect a triangle by a line drawn from a given point out- 

 side the triangle. — J. A. Dodson. 



|3t] — Eleven school-girls went for a walk every day, and were 

 each offered a bouquet every day so long as a different six presented 

 themselves to the giver, and provided also that no ^ire girls ever 



found themselves in the same grojp twije. Ou how inoi,y dayi 

 could they so arrange them«c!vc«? — G. H. T. Bates. 



[22] — Tiic equations are — 



(") 



b X y 



(x — a, v — b, obvijusly siiuuii. loitt.) 

 (li) 2x^/i'»a'■^2x^'x'+ b'-o'-b' 

 May I suggest another astronomical problem ? A lunation 

 2953059 days, and jieriod of the sideral revolution of the moou'a 

 node is C79U 5 days ; show th:it after 1 155b doys eclipses may be 

 expected to recur in an invariable order. — Klevebte. 

 [22u] — Con-ecting tho obrious migprintj, we have 



•'.:+f-a■^b (i) 



a b 



?M:'-«-f6 (ii) 



j; !/ 

 One solution is obviously i — a, i/ — b. But proceeding secundum 

 arlem, i.e., finding value of .t from (ii), substituting in (i), and 

 reducing, we get 



{a + b)y*--2bh/-3abh/ + 2b'{a + h)ij-b'=0 

 As !/ = b is a known root, it follows that y — b is a factor of i 

 equation. Diriding, then, by ;/ — b, 



(o■^b) y'-^(a^-b») ./-(2a(<'-^ b') ,j-tb* = 0. 

 On trial, ij — b is found to be a factor of the new equation, and we 

 get (« + '') y' ■*■ 2aby — h' = 0, a qnadj-at ic, 



whence 



y^a + b^'"^"^"'*'^^'*''''^' 

 besides the previously obtsiined values b, and b. 

 Then, by symmetry, the values of x are 



a, and a, and r( -1+ ^a'l-ab + b') 



a-¥ b^ ' 



V. .). Birr. 



I should be glad of a better solution of the following than 1 have 

 been able to devise. 



y X 

 ,^42 13 



X y 



Science and Art Department, May, 1881. 



In my hands it becomes, by substitution, from ^i) in (ii) 

 .,"_tvkr''-Hl623.c" — 21 952:=0, which, treated as a cnbic, and solved 

 bv Caidan's rule, gives x = 4, and !/ = 2, besides imjigi nary values. 

 But 1 think there must be a shorter cut. — F. J. Butt. 



[.v"-e3y'= -" ; Zt'+y-= — ; whence (x -f y)^ = 21C, x-hy = 6. 



Then 3 (6-y)*-h !/2=---i 3(6-!/)'y -f i/' = 104, giving (y-3)'=-l; 

 i/ = 2; .T = 4, with imaginary roots. — Ed.] 



" [28] 



a^ X g-Hx — 



^a + >/a + x Va— v/o— X 



rationalise denom., and we get 



(o -H x) = + (o - .r) - - 3x v/a 



square (a-)-.r)'-t (a -j)^-i-2 (a=— x') = =9o x' 



i.e. 2a'-3ax»=-2(a»-x»)^ 



square 4 a' -r 9 a-x* — 12 a*i? 



= 4<l«-12a^r■•-^ 12a'x*- 4x» 

 i.e. 4x"'=3n-x', from which we get 



x = 0, or ± v/|a 

 [Similarly solved by Yarletoxia-n and others.] 



W. N. W. 



Om- KUjiSt Column. 



THE GAME IN No. Xlll. 



YOUU correspondent. "Five of Clubs," in his observations ou 

 Z'a pluy (p. 2Si), approves of iTs holding up the turn-up 

 card, and playing a higher one to the adversary's lead of trumps 

 because he thoi-cby gives information to his partner. Now I submit 

 tluU on an adverse lead of trumps, the right rule is to give the ad- 



