270 



KNO\VLEDGE 



[May 4, 1883, 



(9uv iWatt)cmatiraI Column* 



IN our last, wo found tho area we firet set oat to find — viz., 

 A PKN (in Fig. 2) — by an application of tho integral 

 calculus; though wo should have failed it' we had not known that 



A 



= Iogj;, and therefore if we had not first had the particular 



function called a logarithm invented for us, and its qualities, value, 

 and so forth, ascertained. 



But wo want also, now, to learn what is tho integral of ,/j>- — a-dj--. 

 We can form a pretty shrewd guess from the value we have 

 already obtained for the area A P K N' (Fig. 3). If we write x for 

 «! in that expression, and differentiate with respect to ,r, we find 

 that that expression is the integral we want. However, we prefer 

 to work it out proper ly. And noting its resemblance in form to the 

 integral of ^a' — x" we are led naturally to deal with it in the 

 same way. It would, in fact, be obviously the correct thing at this 

 stage to see if the short method given for finding that integral may 

 not be applied directly to this one. Let ua see. Integrating by 

 parts 



/"/-^; ,, I r x'a.x 



■J J Vai^—g? 



J J ^x'^ — a^ J '</xf — a? J ■Jx^-ar 



•■• adding, 2/ v/.^^^x^av/^IT^-/-?^ (1) 

 •' J Vx'-d' 



Hero we find ourselves stopped for want of knowledge as to the 

 dx ix 



integral of , ,. In tho other ease we had, instead, , — ■ — 



vi'_H '■'a'—x-, 



and wo knew tho integral of this to be sin ~' ( - 1 ; but wo have not 



dT 



yet dotormined tho integral of , How shall we set to work 



va;'— o' 



to find it ? Manifestly our proper course is to take advantage of 

 what we have already learned, viz., t hat the part of the integral w© 

 yet have to find involves x + Vx' — a'. This suggests our adopting- 

 th e met hod of substitution and putting i+ '^a5' — a-' = i, — that is 



= log : = log («+ •^x'—a') 



and thus we are able to complete the desired integration (1), anil 

 obtain from (1) 



It will be found that 

 integrated. 



/ 2 2 and v^i'+a- may be similarly 

 (To 6e continued.) 



AN EQUATION. 



H. E. P. C. gives the following simultaneons for solution : — 



11 



~ + — =ft x-~ir=h 



X y -^ 



find :c and y in terms of a and h. It was given him by the late 

 Mr. Bidder. 



[If we consider the conies represented by these two equations, we 

 shall see that they cannot possibly be made to give a quadratic 

 equation in x or y. — E. P.] 



(Bttv 33236 is(t Column* 



By "Five op Clubs." 



"DO YOU PLAY WHIST?" 



IT is amusing to compare the answers given to this question witV 

 the results observed when the game has fairly begun. " Do 

 you play Whist ? " "Certainly! I have played Whist for years,] 

 and I flatter myself I know something about it by this time." The' 

 g.ime begins, and you find the gentleman who has answered so con- 

 fidently knows simply nothing about the game beyond the rules for 

 following suit, counting honours, and so forth, which a beginner is 

 taught in the first ten minutes of his acquaintance with Whist. 

 He not only has no idea of Whist as a game in which each player 

 has a partner, but he does not even know how to play his 

 own hand. He leads out every winning card, weakens his 

 trumps recklessly in rnffing, when — if he knew anything of the 

 game — he would see that by leading trumps, or at any rate reserving 

 his force in trumjjs, he might bring in a long suit. He, perhaps, 

 has just so much thought of his partner as to return whatever suit 

 his partner may have led. A'ery likely he does this when it is his 

 clear duty to show his own suit, or when it should be obvious from 

 tho play that his partner has led from weakness. Or, again, he 

 may so far think of his partner as to force him whenever he gets the 

 chance, though as often as not forcing means disarming. 



Another tells you he plays Whist well who has indeed an idea of 

 the general principles on which sound play should depend, but 

 knows none of the details essential to the application of these prin- 

 ciples in a practical way. He knows, for instance, that when yon 

 lead trumps you generally want trumps exhausted ; but he imagines 

 he does enough in helping you to this end when he returns your 

 trump lead. He is perplexed and aggrieved when you tell him that 

 by returning the wrong card he has utterly foiled all your plans. 

 Thus, holding Ace, Queen, nine, and two of trumps, you lead, let us 

 say, the two, on which fell seven, King, and five; he returns tho 



