Dec. 8, 1882.] 



♦ KNOWLEDGE ♦ 



459 



thicker wire, but with the same core you would have 

 fewer turns, and coiisciiuontly loss E.3I.F. On the other hand, an 

 increase in the size of tlic armature would necessitate a propor- 

 tionate enlargement of all parts. It is evident that if the resist- 

 ance is reduced (the K.M.F. being kept constant) the current- 

 strength must bo increased. 3. Not efficiently or economically. 

 The armature should be wound in sections to produce a current of 

 such proportions. Have you the convenience for making a 

 Gramme? — W. G. Woolco.mbe. 1. The length depends upon the 

 kind of spark you require. About 5 in. would he a useful length. 

 2. The bobbin ends 3 in. in diameter. 3. Core (a bundle of soft 

 iron wire), Iialf-an-inch in diameter. -1. Paper, well gummed. ■>. 

 iioswoiid would answer very -well. 6. Any good text-book on 



<Pur iflatt)fmatual Column, 



EASY LESSONS IN TUB DIFFERENTIAL CALCULUS. 



HERE are a few simple examples of the proce.ss of integration 

 as explained in the last lesson : — 

 OPB is a parabola, OX its a^is, the equation to the imrabola 

 heinij y' = inx. To determine the area B M, when OM=b. 



N 11 



M X 



Draw ordinatcs P N, Q ,i close together. Let O N = .'-, N « = 

 Kectangle P n = ]iCiX. Then, since the area is the sum of all i 

 rectangles as P n between O and B M when they are made 

 definitelvthin, we have area O B M = /" tj dx 



OBM=/'' 



We have to lind out what expression that is which has y'lnx for its 

 differential coefficient. We may write Hii>; \'//' . /'. amX we know 

 that in differentiating u.' raised to any jilwii. ilc index sinks by 

 unity and appears also as a factor in tli.' (lilVcn n!i;d coefficient. 



Therefore .xi must be the differenti:il c lliricni nf .' multiplied 



by a factor which is altered iuto uniiy when multiplied by 

 _- : that is to say, the factor must be ". Ilence 



/ 



v/mj; dx= s/m ■ g,j;i + a constant. 

 (The student should differentiate this expression to make sure that 

 its differential coefficient is v/'nr). Giving to x first the value (■, 

 and then the value 0, we lind 



areaOBM^ti:^ +C-(0 + C) = -'^"'''' 



or, since O M = t, and B M -^ -/mh ; area O B M = ^ roct. BO. 



Of course, there is no occasion in practice for all that has bei n 



hero written out. 



Take another case, to show how such problems are dealt with : — 

 Required the volume of theriqht cone jirodaced by the revolution of 



right triangle 13 M around M, il^a, DM=h. 



Put ()K = i:, 1 

 revolution of rec 



Kfl' M X 



Then 1- K, the v„ 

 ;nd 0.\- = T (P K)' 



oducod by the 



.*. reiiuired volume =^LL/ x'dx 

 Now obviously the expression of which *' is the differentinl co- 

 efficient is -x^-H const. (Differentiate, and see.) Hence 



volume of cone = a' = -t:oX? 



uppose tho following problem sot : — 

 In the cone last considered the density of successive tiincirclaxich 

 as the one formed by the revolution of Pn, varies as the square of OK; 

 required the mass of the cone formed ty the revolution o/ B3f. 



Proceeding as before, we find mass produced by refolulioaof V « 



around X = t(-) ''A-x-x p.x", where p is the dcnsily ak a sait of 



distance fiuiii O along O X. Hence in this case 



Mass 



OV:"'--'iO'" 



_Tr puV 



These examiiles are only intended to give an idea of tie »ee of 

 integration ; we shall consider later the principles by irhick anas. 

 volumes, masses, lengths of arcs, ic, are determined. lot rt. 

 suffice, in this preliminary view, to note how problems wiici, dealt 

 with geometrically, would require more or less of skill or artiiicr, 

 can be dealt with easily and systematically by integration. 



F 



<!^ur WBWt Column* 



By " Five of Clubs." 



PLAY roUIlTII HAND. 

 (Continued from page 424.) 

 PiST, when late in the game you have the King card ai>d u 

 small one, and the play shows that, though led by yonr left- 

 hand adversary, the suit is your partner's, the remaining cards in 

 your hands being all losing ones. If in this case yon let jonr 

 partner's card win, yon arc obliged to win the next trick in the 

 suit and lead a losing card. But if you take the trick with the 

 King card and lead the small one, you are leading through strength 

 11 1> to weakness, and your partner may finesse deeply, and perhaps 

 make all the tricks in tho suit. Usually the case occiirs in the first 

 round of the suit ; but it may also happen in the second. Thus, 

 snppose a suit originally led by your partner from Knave, Nine, and 

 three small ones, you holding King, Queen, and one small one, jilay 

 the Queen, and fourth in hand takes tho trick with Arc. I.»tcr 

 011 (trumps being out) tho latter — your adversarj- on the left- 

 leads a small one (having held originally Ace, Ten, Eight, and :i 

 small one). Your partner plays the Seven, third hand a Iooei- 

 card. If you play the small one, and your partner leads tho fnit 

 again, your King makes, but you have to lead a losing card, and the 

 rest of tho tricks probably go to the adversaries. But if jon iakp 

 the trick with your King and lead the small one, yoax partiu r 

 makes tlxrco more tricks in the suit. 



The second case is one in which you mnst let the adrcTsarics 

 take the trick. When you hold the best, fourth best, and small 

 card of a suit, and a second best is led by yonr left-hand 

 adversary, who also holds tho third and fifth best, yoia nmst pass 

 tho trick. If you win it you must lead through his tcnaco and hwc 

 tho other two tricks; if you pass it, he must lead v^i to your 

 tcnaco, and you win tho other two tricks. 



When, fourth in hand, you have won a trick very rajilr, it is eJtcn 

 good to return your enemy's suit ; for the original leader matt then 

 lilay as if third in hand, hoping for no support from his partner. 

 In trumps this is not safe, however. Even if thinl hand is really 

 as weak as ho seems, you play tho enemy's game by continnii^ tbi- 

 suit. But in trumps it is always possible that a winning card may 

 be kept back to sujiport more effectively later a strong game of th<^ 



4/l'r Drfdtn. 

 * Three Pens fop three esitential virtues famed, 

 Tlie ricku-ick, Ort, mi Wantrlty were named. 

 The ttrat in flexibility surpajiaed, 

 In ease the next, in elef^anee the last. 

 These points united with altractioiis new, 

 Uave yielded other boons, the PhattoH and Jlindo:' 



Sample Box, with • 



I the kinds. Is. Id. )>] 



Post. 



' Let those write now who never \ 

 .\iid those who always wrote now write the more."— OJua 



PalfDl'ri of Pe»« an,l FrnXoldirt. 

 MACNIVEN ft CAMERON. 33, BuinrenasT, EDi»Br««». 



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MlJXSTY'S GOVBHX 



