July 4, 1884.] 



♦ KNOWi^EDGE ♦ 



19 



50. Tho following examples illustrate the method of drawing the 

 lines represented by given polar equations. 



^ 



Take first the equation 



2rcos e + 3r sin 0-6 = 

 Put 6 = 0, giving 2r - 6 = 



That is r = 3 



(i) 



Again put 



9=^,giving3r-G = 



That is r=2 



Thus the given equation represents the line AB in the figure, in 

 which O A = 3, and B = 2, B being drawn at right angles to the 

 initial line X. 



51. The following method is often more conveniently applicable. 



In (i) put as before 6 = 0, giving r = 3; that is, determining tho 

 point A in which the given line cuts the initial line. Xow since 



-G 



r = 



2 cos y + 3 sin 6 



it is clear that if such a value be given to 6 that 



2 cose + 3 sin 6=0. 

 In other words, if we take 



2 

 tan 6= —5 



r becomes infinitely great. Hence if O C be drawn from inclined 



2 

 to X at an angle whose trigonometrical tangent =_-, then 



O C must be parallel to the line represented by (i). Thus we 

 must take A = 3 and then draw through A the line DAB parallel 

 to OC. 



Take as another illustration the equation 



5r cos 9 — 2r siu 6-10 = 

 Here 6 = gives r = 2, and to 



make r infinite we must take 

 5 cos 6-2 sin = 

 „ 5 

 that is, tan 6 = ^ 



Thus if we take 0A = 2 and 

 draw A B inclined to O X at 

 an angle B A X whose tangent 



,8 



is-, AB 

 2 



/ 



/ 



is the line repre- 



.sented by the given equation. 



(To be continued.) 



EASY KIDEES ON EUCLID'S FIRST BOOK. 



With Suggestions. 



Pkop. 33. 



142. Two straight Hues A B and A C are drawn from a point A ; 

 and two other straight lines D E and D F from a point D. A B is 

 equal and parallel to DE, and AC is equal and parallel to D F. 

 Show that B E is equal and parallel to C F. 



143. If a quadrilateral have two of its sides parallel, and the 

 other two equal but not parallel, any two of its opposite angles are 

 equal to two right angles. 



144. Two equal but not parallel lines make equal angles on the 

 same side of a thii-d line which joins their extremities. Show that 

 the straight line which joins their other extremities shall make 

 equal angles with the two first lines and be parallel to the third. 



145. In the figure to Euc. I. 5, G L drawn perpendicular as to 

 B C produced, is produced to M so that L JI is equal to L G. Show 

 that B L is equal and parallel to F C. 



Prop. 34. 

 14G. The diagonals of a parallelogram bisect each other. 



147. If two straight lines bisect each other, the straight lines 

 joining their extremities form a parallelogram. 



148. No two straight lines dravm from the extremity of the base 

 of a triangle to the opposite sides can possibly bisect each other. 



(To ie continued.) 



4Buv mxin^t Column. 



By Five of Clubs. 

 The Hands. 



, f H. K, 7. 

 iC. A, 7. 



• H. Kn, 10, 0, G. 



P. A. Q 

 |i, K, :i. 



4, 3, 2. \ 



.> C. K. 

 1 D. A, Q, 

 (.S. 8, 7, C 



Kn, 9, . 



f II. Q, 8. 



{C. Q, Kn, 10, G, 5, 4, 2. 



S. 10, 9. ■^ 

 D. 7, 4. i 



B 



'.^21 



Z 

 + 

 * 



0^0 



4 



11 



* -f ♦ ♦ 



THE GAME. 



1. -1 leads from his long suit, 

 and properly leads Queen from Q, 

 Kn, 10, <Sc. The suit is esta- 

 blished first round. 



2. B seeing his partner's suit 

 cleared (of course it is not known 

 to B that .4's suit is established}, 

 and having commanding strength 

 in Spades and a King guarded in 

 Diamonds, properly leads trumps. 



3. After this round T remains 

 with command in trumps; jB cannot 

 hold more than two, or he would 

 have led Heart three originally — 

 tlie penultimate. 



4. Z properly leads from length 

 rather than from strength. 



5. A properly discards from his 

 weakest suit. 



G. Y clears out trumps before 

 leading from his long suit. 



7. So far as this round shows, T 

 may not have length and strength 

 in the suit originally led by Z, but 

 may be simply playing to clear Z's 

 suit. 



8. B renouncing shoivs Z how 

 the Diamonds lie. T holds Queen, 

 nine, and five. If Z plays the 

 lowest he will have to take the 

 next round in Diamonds, and either 

 to lead .4's established suit or the 

 suit in which B is presumably 

 strong. But playing out the ten, 

 Z clears his partner's Diamonds, 

 gaining one trick in that suit. 



9. B plays very badly here. The 

 other three tricks are in A B's 

 hands, and all wanted if game is 

 to be saved. But the weakness so 

 many players have for making 

 tricks themselves rather than let 

 their partner make them, causes 

 B to finesse the Queen, instead of 

 taking the trick with the Ace and 

 leading Club to his partner's com_; 

 manding sequence. 



The above hand is from Caven- 

 dish (Notes by Five of Clubs). 

 It illustrates a point which in- 

 experienced players constantly 

 overlook,. — the necessity of getting 

 rid of command of partner's suit 

 when that suit has been esta- 

 blished and can be brought in. 



