Nov. 10, 1882.] 



• KNOWLEDGE 



395 



&UV iflatfjfmatiral Column. 



EAST LESSONS IN THE DIFFERENTIAL CALCULUS. 



By Richasd A. Pkoctor. 



No. XII. 



I PROPOSE now to give two geometrical illustrations of a dif- 

 ferential coefficient, which, when their nature is rightly under- 

 stood, and especially the circumstance that the various values of a 

 fnnction can always be expressed by means of a curve, will be found 

 of great value in indicating the real meaning at once of differen- 

 tiation and integration. 



Let be the origin, X and Y, at right angles to each other, 

 the axes of x and y. 



Then the x we have been dealing with — our independent variable 

 — is measured along O X, and y the dependent variable along O Y. 

 So that when we write !/=/(j'), i-c, y is such and such a function of 

 .T, we may represent any values of x by M, O N along O X, calcu- 

 late the resulting values of y, and set up corresponding lines M P, 

 N Q parallel to O Y. If we suppose this done for all values of x, 

 we get in every case a curve such as is supposed to be shown in 

 part in AP Q, theordinatesas M P, N Q representing the values of y 

 corresponding to the values of x represented by the abscissa^, as 

 O M, N respectively. 



Supposing, then, that when a: = O JI, y = M P, we may take M N to 

 represent a finite increment of .t or Ax, and get N Q for the cor- 

 responding value of y. Draw P L parallel to X, cutting off 

 N L = P II from Q N, this new value of y , — or y + Ay. Then P L = A J: 



andQL=A!/. And what we have represented by -=1 is the ratio 



Ax 

 Q L : P L, or the tangent of the angle Q P L, when P Q is a secant 

 line. If now we imagine N brought nearer and nearer to P II, it is 

 manifest that the secant line PQ draws nearer and nearer in 

 position to the tangent line P T, and the ratio Q L : P L approaches 

 nearer and nearer in value to the ratio T L : P L, the trigonometrical 

 tangent of the angle which the geometrical tangent to the curve 

 A PQ at P makes with the axis of x. Hence the differential co- 

 efficient J^ = tangent of the angle T P L. 



Here wo have at once an illustration of 

 of a differential coefficient and a useful : 



ential calculus. The reader of treatises on elementary plain con- 

 dinate geometry knows how important a process is the determination 



of the equation to the tangent at a given point of a curve, and 

 how cumbrous is the method which has to bo employed for its 

 determination by elementary methods. With the differential 

 calculus the process is simplicity itself. 



Thus, suppose A P Q is a part of the rectangular hyperbola whoso 

 equation is 



and that we require the equation to the tangent P T at a point P, 

 whose ordinates are O M = a; , and P M = i/ , . We have 



1/ = - ; ^ -, whence tangent P T X — 



.r dx I- x,2 



and the equation to P T is, therefore. 



or, x,y + y^x=2x^y^'=2a- 



Again, the equation to the ellipse with origin at centre and major 



axis as axis of ,r, is - -(- ^ = 1 



a' b- 



•' <y""--''' d:c «^?ri? 



^\ Lcrefure the equation to the tangent at a point a-,, y,, on the 

 curve is t/-y, ^ _h x, b'x, 



or, V 1 y + ^'x 1 X = ahj j ^ -^ fc'j; ^ » := a't" 



0m 212ai)ist Column. 



By "Five of Clubs." 



DEAR FIVE, — I have just been looking over your notes on the 

 play of the illustrative game (p. 367, No. 52) which I sent to 

 you two weeks ago. It seems to me that your remarks on Z's play 

 may seem to him rather im compromising. Suppose he should be a 

 reader of Knowledge and recognise the game. Might it not seem 

 to him that, as a visitor, he should have escaped comment ? — 

 Faithfully yours, EnrroB. 



Dear Editor, — Impossible to annotate a game fairly without 

 noting errors of play. But are you in earnest ? Who was it sent me 

 a question two weeks ago, relating to play at the very same meeting 

 of your club, and telUng me to "be as severe as I pleased in 

 pointing out error," &c. r Such mistakes are made in moments of 

 carelessness by the best Whist players, and no true lover of the 

 game objects to comments on them, if sound. If unsound, they can 

 always be refuted. — Yours faithfully, Five of Clubs. 



Dear Five. — Well, yon certainly responded plainly enough to my 

 request for plain speaking, and rapped P. A. R. (R. A. P. inverted) 



at p. 347, pretty severely, 

 and well-delivered thrust. 

 editor's tastes. I agree i 

 evening ^rithout laying lii 

 would pain nie veiy mucli i 

 under your comments. I v 



:sut not every one enjoys as i do a fair 

 liCt us hope our visitor shares your 

 ill y(Mi tlirit no player ever passes an 

 1^' If (>! ■ Ti i(. some criticism. Still, it 

 :iiiy 1.11, ', f,'c<lii)gs wore at all to suffer 

 11 oiie ii 1 i.cii recall the game in which 

 I blundered so egregiously, and you shall criticise that if you like. — 

 Faithfully yours, B. A. Proctoe. 



The Whist Editor of the Australasian has been good enough, at 

 our antipodes, to watch tho progress of our Whist Column, in 

 the hope that we had taken np the mantle of tho "Westminster 

 Papers ; " and now ho is disappointed that wo have not adorned 

 ourselves with that garment. Considering wo very definitely indi- 

 cated a quite different purpose, this is not very much to bo wondered 

 at. We proposed from the beginning, or rather at the beginning, 

 to describe the elementary principles of Whist, such as tho leads, 

 play second, third, and fourth in liand, the return of tho lead, and 

 such matters, showing how inferences may be formed as the play 

 progresses, and the importance of making and remembering such 

 inferences, leaving to a later stage tho jirinciples which guide tho 

 general conduct of the hand and may lend the practised player to 

 depart as tho game progresses from tho niei-o conventional lino of 

 book jilay. In a word, we proposed (iind still i>ropose, for much 

 «{ (his initial matter remains still to bo written) to teach the 

 lii'L'iniior those rules which every Whist player ought to know, and 

 wl.icli beginners ought, for tho "most part, to follow closely. The 

 ■ Westminster Papers" had a quite different purpose. 



In carrying out our plan, we had occasion in the number for 

 June 30 (Vol. II., p. SI), to give a game si^ocially to show how 



