394 



KNONA^LEDGE • 



[Maucii .;, 1882, 



bilnd iiPTornI portionii of tho inHtrnmonl,ilnd raontioned tlio variiniH 

 iliflimiltii'ii likiily to Iw oncoiintorod. and tho moniiii proviiltMi fur 

 dcaliiiK with thoin. 



STHrNOTii or Matbkui.i. — At tho hito fair of tho Mus»achii»illH 

 Charitable Moohuiiic A«Horintion, at Uowlmi, oxiimploH won- Hhuwu 

 of tostii of nmtvrialB inndo by tho machino lately orertcd in l\f 

 United Slates (lovorninont Arsenal, at Watortown, for the provini; 

 of BtnicturoH of full working dimcnnionM. A stool wiro cable. 1 i| 

 inehos diameter, was shown, which had withstood a pull of 7o tons, 

 when the fasteninj.'s by which it was hold (fnve way, allhouKh t ho 

 cable itself remained sound. A hammered iron bar, 5 inchi-s in 

 diameter, was shown to have concealed a crystalline formation of 

 tho fibres, and it conseipicntly parted with a loud report under a 

 strain of nearly 723.01H) lb., or 3t;.900lb. to the 8<iuare inch. A 

 smaller wrouRht-iron bar drew down and broke with a fibrous 

 structure under a pull of 51,310 lb. per sipiaro inch. Some jjine- 

 wood columns wcro also shown which hnd boon tested by comiins- 

 gion. Tho first of these, oriRinally 1:2 foot long, yielded iit a 

 pressure much bolow its ostimatcil strength, in coiisof|Uonco of a 

 largo knot in the side, which acted as a comparatively incom]>re8- 

 siblo wedge. Another column was a spar 1 a feet long, 7J inch 

 butt, and Gl inch top. This stick was a perfect sample, and gave 

 way by splintering at its smaller end. A seasoned hard pine 

 ginior, 11 inches scpiare and 10 feet long, bore a load of 751,000 lb. 

 — Scientific American. 



(!^ur iWatbcmatiral Column. 



Find the area intercepted between n litjperholu, an asijiniitotf, n;in 

 tiuo ordinates parallel to the other aS'jMptote. 



M^FT 



Let OAB, OHK be the .isymptotcs ; AD, BC^bawu parallel to t)K 

 to meet the curve in D,C. In OAB take OR greater than OA, bnt 



so that AR is very small ; take points S,T L,M . . . Ac, so that 



OA: OR::OR : OS::OS : OT OL : OM ,and let B'be the 



nearest of such points : to B, so that B'B is small' Suppose that 



OB' is thus divided into n parte AR, RS, ST, &c , and let 



LMbethe (r + l)thsueh part. Draw LP, MQ parallel to OK to 

 meet the hyperbola in P and Q, and draw DH, PQ. Q/F, C'E parallel 

 to OB, QF meeting PL in /. Then, since OA : OR : : OR : OS : :0S : 



OT, &c , and that LM is the (r + l)th of the parts AR, RS, ST, 



&c., i.e., M is the (r+ l)th of the points : of division R,S,T, &c. 



OA : OM::OA'+' .-OR-^-' (i) 



and sinularly OA : OL : : O A' : OR' 



::0A'+' : OA.OR'. 

 Hence OA : LM : lOA'*' : OR'.AR 



But DA:MQ::0M OA sincelOA . AD = OM.MQ 



::0R'^-' : OA-^' from (i) 

 Thus OA . AD : LM . MQ : : OR : AR 



or parallelogram OD : parallelogram LA:: OR : AR 



t.e, all the parallclognims inscribed as LQ is are equal : thus paral- 

 lelogram OD : sum of inscribed parallelograms:: OB : ;i AR 

 But sum of ])arallelogram8 = ADC'B' = ADCB ultimately, when AR 

 is taken indefinitely small. 



llcnco ADCB : parallelogram 0D::7i. AR : OR (ii) 



But ultimately OA" ; OR" : :0A : OB, i.c./2By=|2| or » log. ^ 



, OB 



= log. — 



• •* OA 



, OB , OB , OB 



log. log. log. 



f)A OA " OA 



! TiTt "1 t 1 + AR "> " AR 



l.'i.'. — log. • — ; ~ 

 OA '. OA> OA 



OA 



OB 



ultimately log. 

 \R ' OA 



f)B 



ADCB : parallelogram OD::OA log. LIU ; oK or (lA 



_ since OA- OK ultimately 



Kditok. 



[25, p. 307] — There ap|)earB to I)e an error in your solution of this 

 question. You have taken the number of months as 20, instead ot 



120, £- for &L- 

 5 6 



The following appears to be a solution : — If r=-the rate of 

 interest per cent, per month, then r has to bo found from the 

 equation — 



1-a + r)-" 



110 = 100. 



1-(1 + .)- 



It will be found by logarithms that •005918 is a very near value of r 

 in this equation. Hence £'5918, or lis. lOd. per cent, per month is 

 the rate of interest realised. — J. McGowAX. 



[Query No. 200, p. 278]— Leases. — W. Cahill's query in No. 15, 

 in connection with James Gregg's previous query :■ — 

 Assuming the interest to be 5 per cent., the <; P'Hmsl" 



premium of £1,050 paid at commencement ' " 



would amount at the end of 14 years to 2 I'T- 



An annuitv of £250 would amount at the end of 



10 years to 3,\H 17:5 



The interest on same for 4 years (to end of 



the 14) C77 657 



An annuity of £300 would amount at the end of 



4 years to 1,293 038 



Total value of all at end of 14 years 7,194 096 



The present value of this amount being 3.633 5 



Which would buy an annuity or the lease at a 



pc]ipercorn rent of 367 07-5 



Deducting the £300 rent paid each of the last 4 



years, the premium to bo paid at the end of 



lU years is the present value of 4 years' 



riimuity of £67075, which amounts to 237 845 



— .] . W. The Answer. 



©ur ©abi'St Column. 



By " Five of Clubs. ' 



AN ILLUSTRATIVE GAME. 



CLAY, in his chamiing little treatise on " Short Whist," givot 

 the following interesting instance of the danger of continning 

 a forward game, when early indications show that the promise of i 

 great score w^as fallacious. (In passing, one may note that in cases 

 such as this information of weakness may prove exceedingly nsefol 

 to tho stronger partner, by showing him the necessity of caution ; it 

 is in this respect that tho ordinary game differs from dummy play,, 

 when the danger is indicated at once. Some dummy players an*; 

 apt to overlook this negative advantage of intimation of weakness' 

 in jiartnor's linnd, and to consider only the more obvious positive 

 .idvantage which necessarily accrues to the adversaries): — 



" I dealt," says Clay, " and tm-ned up a Queen, along with which I 

 held two small trumps. My partner — nor was he a bad player — 

 held tho Ace and four of the smallest trumps, and. so to speak, the 

 whole of another suit. With this strength, assisted by my Queen, 

 lie promised himself, reasonably enough, a great .score, if not the 

 whole game. Bnt the first two tricks showed him that he would he 

 overt nimped. He should have submitted to this, and as it happened 

 he could have made a good score, but he was unable to dismiss the 

 idea of a strong attack. He trumped the second trick with his Ace, 

 led a tnimp, — and we made no other trick. Thus with Ace, Qneen, 



