Feb. 17, 1882.] 



KNOWLEDGE 



349 



with onx original pliin ; and that, as that plan involverl a promise, 

 we must adopt in future a different line. We cannot further find 

 I space for such prviblems as occnr in ordinary mathematical reading, 

 ' except when illustrating general principles. Our mathematical 

 i column must not degenerate into a puzzledom corner. — En. 

 ' A correspondent (X. XL.) asks for a demonstration of a property 

 of the conic sections to which we referred a short time back, viz., 

 that if a sphere enclosed in a cone (like a ball in a conical cup) 

 touches the plane of section, the point of contact between the 

 sphere and the plane is a focus of the conic section. We have pre- 

 pared three diagrams corresponding to the case of (1) elli[>se, 

 (2) hyperbola, (3) parabola, and ^vill give these next week, with 

 a demonstration which seems to ns of interest, as probably the 

 simplest proof connecting the fundamental property of the ellipse, 

 1 parabola, and hyperbola (relation between distances from focus and 

 I directrix), and the fact that the curves possessing tliat property 

 are sections of the cone. — Ed. 



MATHEMATICAL QUERIES. 



[35] — Value of Lease. — Given, H} years repairing lease ; 

 rent, £15 ; ground-rent, £i ; present rental value, £32. Kequired, 

 present worth of lease to make 5 per cent, interest. — James Gbegg. 



[36] — Can you, or any of your readers, tell me how to obtain the 

 general term in the expansion of (Oj -^ a., +a^+ &c.)', n being whole 

 or fractional, positive or negative ?— Cartesian. 



[We should deal with the problem somewhat on this wise: — Let 

 any expression of the form a,„ + a,„+j -^ a„^.; + Ac., = am : also in the 

 expansion of (o„, -^ a,.+i) ' take the (r,„ + 1) th term for general 

 term, and put p — / „ — r„. Then 



. ^ , X rn(n-l)...(n—r,+l) "l 



(o,-f«t,-K73 + &c.)" = (a,-iao)" =S [-^ ^g3_ ^ ' a/ia/. J 



|r, 



finally (a, -i- «„ + a, + &c.)" 



^j, p „(,v-l)...(„-,.,+l) .^ ^^ 



Where r., i-j, >■_,, &c., are positive whole numbers, and 

 n = r, -h r„ + )-3 + r^ + &c., 



If n is a positive whole number, we may conveniently interchange 

 and Tj in the first part of the process (the distinction being only 



ntroduced because if n is not a positive integer, neither is r,). 



Ve thus obtain the convenient formula 

 , , ^m (n-1) .. (n-r, -t-l) 



(aj'. = (a3-^a,)' 



(a,)'3=(njH-ai)' 

 ic, 



lis. 



3'3a/3J 



ic, 



!'■> 



#iu- SMbiSt Column. 



WUIST PROBLEM, No. 1. 

 In this problem B holds the following hand ; — 



Spades. — Ten, nine, six, five. (Tramps.) 

 Hearts. — Ace, Queen, four, two. 

 Diamonds. — Queen, six. 

 Clubs. — Ace, ten, eight, 

 ad the four first tricks are as follows, the underlined card winnin" 

 ■ick, and card below leading next : — 



After these four tricks have been played B is able to place every 

 card, supposing tliat all the players have followed the usual roles 

 for play. 



No one has solved this problem correctly. Fifteen solutions sent. 

 We note that what we have hitherto said about whist leads does 

 not quite suffice for the solution of this problem, though it help 

 towards it. It is necessary to supplement the rules for lead, 

 however, with only two general rules, one for second, the other for 

 third player, to give the solution. These are first that second 

 player, if he has a sequence of two high cards and one small one, 

 plays the lowest of the sequence second hand on a small card led ; 

 secondly, that third in hand plays highest if he has any card higher 

 than (and not in sequence with) his partner's lead, and no sound 

 finesse open to him, but otherwise plays his lowest. 



First Ti-ick.—A has led the lowest from four at least (it should 

 have been noticed that the inventor of this hand did not accept the 

 rule for penultimate lead). Since two is not in .-I's hand, nor in 

 Z'b, for Z's lead third hand shows he was not signalling for trumps 

 and B has it not himself, it must lie with }'. Hut no other small 

 card can be in i'shand, who would only play Knave, having the two 

 if he held Queen, Knave, two, ajid no more. Hence four and five lie 

 with Z, and no more, for A must have four Clnbs. Thus the Clubs 

 were originally distributed as follows :— With Y, Queen, Knave 

 two ; withZ, five, four, three ; with B, Ace. ten, eight ; and the rest,' 

 viz.. King, nine, seven, and six with A. 



Second Trick.~A has no Hearts above ten, and his play of nine 

 shows he has none lower. Hence, A only holds Hearts nine. As T 

 plays the five, he does not hold the three (he had not begun a signal 

 m first round, as B knows, holding Clubs U-d in his own hand). 

 Hence, Hearts three must be held by Z, and as he played ten, having 

 the three, he must have the Knave, but no others. Hence, the 

 Hearts lay originally as follows : — 



With A, the nine; with Z, Knave, ten, three; with B, Ace, 

 Queen, four, two ; and the rest, viz., King, eight, seven, six, and five' 

 with Y. 



Third Jrick. Diamonds four is the lowest of four at least. A has no 

 card below the eight, hence the two and three must be with I" as^ 

 IS certainly not signalling. We know also that A has not five trumps, 

 or he would have begun with one ; hence, as he had originally four 

 Clubs, one Heart, and fewer than five trumps, he must have more 

 than three Diamonds. Since eight is his lowest and Z has led from 

 four at least, B having Queen, six, and Y Knave, tlu-ee, two, it follows 

 that Z must have held seven, five, four, and either Ace or King, 

 showing that A must have had eight, nine, ten, and either Ace or 

 King.^ But A' a first lead shows that A must have the Ace and not 

 the King, for he would not have led Clubs from six, seven, nine 

 King, if he had had eight, nine, ten. King of Diamonds ; thou'^h' 

 foUowing Clay's rule, he would have led a Club if holdins eight,' 

 nine, ten, Ace of Diamonds, reserving the Ace-headed long'suit to 



get in with later. Thus the Diamonds lay originally as follow : 



With y, Knave, tlu-ee, two ; with A, eight, nine, ten, Ace ; with 

 B, Queen, six ; and the rest, viz., King, seven, five, four, with Z 



Fourth Trick. —B knows already that A holds four Spades- Y 

 two Spades ; and Z, three. As Z plays the seven, tlio onlv card^ 

 left which can make up his remaining two are the eight, the"Queen, 

 and the Kmg. He cannot have both Queen and King, or he would 

 have played the Queen. He must have, then, either eight Queen or 

 eight King. But if he had the Queen, King would lie with A, and 

 A would not have finessed the Knave holding King, Knave, and two 

 others. Therefore Z held King, eight, seven. }''s other card 

 must be a small one, and Spades were originallv distributed as 

 follows : — 



Z,— King, eight, seven; B,— ten, nine, six, five; F,— Ace, two (or 

 three, or four) ; and the rest, viz., Queen, Knave, four, three (or four 

 two, or three, two) with A. ' 



The doubt as to the actual value of the small spade in Y's hand 

 can hardly be said to affect the statement that Z knows the position 

 of every card in the pack, for the two, three, and four, are in this 

 case of practically equal value. 



We would now leave our whist readers to explain why B led 

 trumps fourth round, when, with his knowledge of the position of 

 cards he might, one would say, lead his only remaining diamond, 

 through Z'.< King, enabling A to make the trick with the nine. 



G. Thompson-. B's lead second trick is correct. It is unfortn- 

 iinte having to load from a tcnace suit; but it is better than 

 decemng partn.r. Returning partner's suit at once means, "I 

 have no strong suit."— H. P. YARMouxn. Your method of dealing 

 with the problem discussed by the Editor at p. 301 (letter 259) is 

 incorrect. Do you not see that in five cuttings, according to your 

 method, A would possess fifteen chances out of thirteen, which is 

 absurd?— Gr-ii.atio.v. The lead of King followed by Queen from 

 Ace, King, Queen, Ac, should certainly have been added (it is indi- 

 cated at p. 259) ; but not " Ace followed by ten from Ace, Queen 



