120 



♦ KNOWLEDGE • 



[JuLv 14, 1882. 



^ur ifiattjnnatiral Column. 



EASY LESSONS IX THE DIFFEKENTIAL CALCULUS. 



No. in. 



Bv THE Editok. 



TUE plan wc adopted in Lesson L to obtain the differential 

 co-ffficicnt of the expression i gf with respect to the variable 

 ', will give na the differential co-efficient of many other variable 

 expressions. 



It may be well to trj' one other case, giving also an illustration 

 of the valae of such processts, before proceeding to obtain the 

 differential co-efficients of ^•arious familiar functions, 

 sc, for instance, that 



Increase x by Ax, 



(i) 



I y by i« ; then 



so that, subslractiiig (i.) from (ii.), 



AJ: = aA!/— 2i/A!/- (At/)- 

 and— f =a-2i/— Ay. 

 Ay 

 Now make Ax and Ay inOnitely small, calling the 

 Then wo get 



(ii.) 



dx- and ihj. 



-j- = a-2y-((y = a-2y. 

 dy 



This process would be verj- cumbrous if applied to complex ex- 

 pressions. Therefore, the first matter considered in treatises on 

 the differential calculus, is tho determination of rules by which a 

 differential co-efficient may be readily obtained. In tho next paper 

 on the subject I shall give sgme of these rules, without dwelling; at 

 any great length on the reasoning by which they aro established. 

 Much of this reasoning, indeed, would bo beyond those for whose 

 special service these papers aro written. The advantage derived 

 from the practical application of tho differential calculus to pro- 

 blems not easily sslved in other ways, will encourage tho student to 

 discuss after awhile the reasoning by which tho rules of tho calculus 

 have been established. The great difficulty has hitherto been that 

 this reasoning, coming before the student has learned tho power of 

 the calcnlu.<>, has, by its length and complexity, prevented many 

 from ptirsuing the study of the subject. 



But even at this st,ngc, it will be well to illustrate the application 

 of the differential calculus. 



Suppose we had this problem given : — 



The Icnn'h '/ the line Ali i» A; v.herc must a j^oint P Ic talien in 



ordir that the rectangle under AP, PB may be as great as poaaible ? 



Since A I' is y, P B is (a-y), and the rectangle under AP, PB 

 (which call i) is y (a- y) ; that is 



x = oy-y'. 

 We want z to be as great as possible. Now the differential co- 

 efficient of z with respect to y, is tho rate at which x increases with 

 increase of y (from to a) ; and so long as z is increasing, x is not 

 as great as possible. We must find then when x ceases to increase, 

 or when its rate of increase (or its differential co-efficient) is reduced 

 to nought. Now we have seen that when 

 z = ay-2y, 

 dz 



ratting this e<iual to nought, we have tho equati( 

 a-2^ = 0. 



~ J that P must bisect AB, in order that the rectangle under AP, 

 i'B may be a maximum. 



Ilerw is another problem : — 



ACita eylin'Jtr ecrutructed to fulfil the condition that its height 

 Ali added to AO, the radiut oj a circular face, it C'lual to a 



:^3 



fittd length A. Required the height of the cylinder in order 

 ill curved rurface nuiy le at great at joatible. 



Pot AB -y; so that AO-o-y. 



Then the curved surface, which call .r, is represented by the rec- 

 tangle under tho height and the circumference of a circular face. 

 That is (representing tho ratio of the circumferenco to tho iliamcter 

 as usual by t), 



z = 27r (a-y)y = 2n-(ay--y>). 

 Here, as before, we must havo tho rate of increase of a with increaso 

 of y (or the differential co-efficient of x) nought. But if wo went 

 through the process for determining the differential co-efficient as 

 above, wo should readily get 



dx 



and tho cquatii 



T(a-2i/) = 



(To be continued.) 



0ut 212abi5t Column* 



By " Five of Clubs." 



DEAR FIVE, — I have been studying Whist einco Knowledge 

 camo out, having foimerly played a good deal, but without 

 much knowledge of tho principles. I see you aro working your 

 way (for us learners) through tho leads, play second and third in 

 hand, return leads, and so forth. But I have a question to ask 

 about matters you have not yet reached — illustrated in tho follow- 

 ing game, which forms the first example in I'ole's lucid treatise on 

 tho " Theory of Whist." Will you kindly givo some notes on tho 

 game. I do not express my own opinion as to particular points of 

 play ; but speaking generally, I may say that it seems to me tho 

 play on both sides (not of all four iilayers, however,) is decidedly 

 bad. — Faithfully yours. Deuce ok Hearts. 



In tho game referred [to by " Deuco of Hearts," as given 

 by Professor Pole, the score, on which in reality tho play 

 would greatly depend, is net given. We assume that it is "love 

 all ; " but if it were A B love, 1' Z three, B's play would be better 

 justified than it is under the assumed actual conditions, because 

 then nothing could save A B (if honours against them) but the 

 possession of such cards, or at any rate such a long suit by A, as B 

 ought, under ordinary conditions, to liold himself, — to justify his 

 signalling from five trumps one honour. On tho other hand, if the 

 score were A B four, Y Z three, B'a jilay would bo about the best 

 he could follow to lose the game. Tho play o( Y Z also would 

 depend much on tho score. Tho game is as follows : — 



The Uand.s. 



1. A loads correctly (see leads). 

 Tho Two not falling, ho knows 

 someone is signalling. B com- 

 mences tho signal. At tho assumed 

 , B plays very badly in signal- 

 ling, though if the lead had been 

 with him ho would havo been right 

 in leading trumps. When a player 

 leads trumps ho says to partner, 

 " I am strong enough to play 

 a fonvard game if you havo 



