170 



• KNOWLEDGE • 



[March 16, 1883, 



not squared tho circle for as; for it Ims not shown the length of tlio 

 eircumffrence, or of the arc AB iu tlii^ particnlar case dealt with. 



It is only because we know the lcn(,'tlis of circular arcs that our 

 result is really a dutcrmiuation of tho area of the sector 



Suppose, now, we wish to determine by means of the integra". 

 calculus the area O A B K (Fig. 2), where A B L is a quadrant. 



We have already determined the area in what precedes ; for. 

 joining B, we know the area of sector A O B, and adding thereto 

 the triangle B K, we have the total area O A B K. 



Thus if radius A = a, O K = x„ then B K = v/a' — Xi^ ; and arc A B 

 = o sin ~'(—\ Hence area O A B K = triangle O B K + sect. A O B 



——Lz. 1 + — sm 'I— t I 



2 2 \o / 



But snppose we wish to determine the area O A B K indepen- 

 iently, by the summation of such rectangular strips as Q M, made 

 in the limit indefinitely narrow. 



We put OM = ir, MN = oa;. Thus we have — 



Eect. Cl'i>l = •/(!:'— X- d X; and area OABK = sum of all such 

 rectangles made infinite in number, and so taken indefinitely thin, 

 oetween A (where x=0) and B K where x = Xi. 



Thus Area A B K 



x'dx. 



Now, to integrate I ^d'—x' dx ^'^ may try the method of integra- 



aon by parts, noticing that unity is the differential coefiicient of x 

 (rith respect to x. Thus — 



y^"-' 



But here we notice that — - 



(1) 



■^a'—x' 

 So that f—i^ = - f^liF^^ix + a- f—l — (2) 



tho first part of which, when written in equation 1, will clearly give 

 OS what can be conveniently taken over to the other side with a 

 positive sign, while the other part is in our table of known integrals 

 at p. 473. Thus, substituting in- (1) from (2) and transposing, we 

 get:— 



/* /- dx f—^ « . _iai 



Or, 



/^ 



Putting for x in this we 



- ar Jx = 



/■ 



:et ; so that 



This, therefore, is the required area O A B K. 



Remembering that this area O A B K can be divided into tho 

 triangle B K and the sector A O B, we have a ready means of 

 rememhering as well as of intcrpreling the important relation 



/ V or— US' dx = 5 + — sin ~' - 



We have given the solution for this integral, as it might bo found 

 out by one seeking it without a knowledge of the bo<jk method. 

 Ilaving found it, wo can conveniently write the solution thna : — 



/ ■v/a'— *• <fx=iv''a' — 1'+ / . - 



y' , /■ o' — i' /■ a'dx r x'dx 



:. adding, 2/X/^r:^dj=j,vV:i'^+ /"-4^^^ 



y/~i i , x^a? — 3? "' . , " 

 Va'-x'da:= T +- sm-" - 

 2 2a 



But the student who really wants to understand what ho is doing 

 should work out such results in some such way as lie would in 

 actual practice, not follow simply the conveniently abridged 

 methods which are formed after a result has been obtained. 



#ur WAii'it Columm 



By " Five op Clubs." 



PLATING TRUMPS- (confiftued /romp. 153). 



I HAVE been careful to consider first the cases in which a back- 

 ward game should be played with regard to trumps, and the 

 book rule about passing doubtful cards neglected; because so many 

 games are lost through a rigid adherence to a rule which is often 

 misunderstood. As a matter of fact, the rule is not properly stated. 

 Strength iu trumps is not a sufiScient reason for refraining from 

 ruffing, if such weakness in plain suits has been shown as to suggest 

 a backward game. The rule should rather run. So long as there is 

 nothing to show that between trumps and plain suits you and your 

 partner can hold your own against the adversaries, refrain from 

 ruffing a doubtful card from a four-card trump suit ; and equally, 

 Under the same conditions, refrain from forcing your partner if, 

 being yourself weak in trumps, yon have reason to think he may 

 be strong. 



But turning now to cases where there is no special reason to play 

 either a backward or a forward game, we see the reason in these — 

 which includes the mnjority of cases — for the book rule. In every 

 Whist hand in which the forces are fairly divided there may be said 

 to be on the average sixteen tricks in the four suits of which only 

 thirteen can possibly be taken ; and thus in every fairly matched 

 Whist hand there are two or three tricks which one or other side 

 will make or lose, according to the skill or good fortune with which 

 the hand is played. Of course, in the play of each suit there is 

 room for skill and good fortune to tell, in tbe finesse and so forth; 

 but we are considering, just now, the total number of tricks to bo 

 made, whether by finesse or otherwise, in the several suits, if each 

 could " tell" to tlie last card in it. Now, as a matter of fact, this, as 

 a rule, is only the case with trumps, though it may happen with any 

 estabhshed suit after trumps are esliausted. In trumps there may 

 be more than the natural number of tricks, because trumps may 

 take tricks singly. 



But a time comes in every well-matched hand of Whist when the 

 question which side will make the most of its long plain suits (by 

 which is to be understood every suit of more than three) depends 

 on the manipulation of trumps and forcing cards. All may then 

 turn on the possession of a trump more or less, on one side or the 

 other. Forcing a hand which holds more than the average number 

 of trumps thus very often means the gain or sacrifice of one trick 

 (according as it is your partner or the adversary who is forced) for 

 two or three tricks. This is so commonly the case in well-matched 

 hands, or hands which have become well matched so far as what 

 remains of them is concerned, that we get almost as standing 

 rule in such cases, Refrain from forcing your partner if you are 

 weak in trumps and he probably strong ; Pass a doubtful card if 

 you hold more than the average number yourself; and Force the 

 adverse strong tramp hand. 



But these rules do not apply to very weak hands, even though 

 there is numerical length in trumps ; nor, on the other hand, do 

 they apply to very strong hands, which can often aSord to ruff and 

 wait, or even to rnff and lead trumps. 



The determination of the proper time for leading trumps in the 

 case of fairly-matched hands not justifying either a signal or an 

 early load of trumps, still remains among the most difficult points 

 of VVhist strategy. It may generally be taken for granted that a 

 lato lead of trumps, unless obviously forced, is to be respected by 



