November 1, 1886.] 



♦ KNOWLEDGE ♦ 



11 



the spaces between the divisions rapidly diminish as shown 

 (no special construction is necessary). Then proceed as fol- 

 lows : Opening a pair of dividers to span ca, keeping the 

 end of one leg at c, mark the point ;/ on bd with the other ; 

 then opening to span b^, and setting the end of one leg at 

 c, mark the point ;/' on en with the other ; do the like, 

 starting with the span cii, getting the point h' ; and so get 

 a series of division marks //', h', k, I' , m', n', and o' on CB ; 

 dividing ad similarly at «' (such that dg' = b,7'), h', k', ifec. 

 Again, opening a compass to span cp, mark the point p on 

 AD, having end of one compass-leg at c, and with span Ap 

 measure off Fp' ; do the like, starting from the span cq, 

 getting the point q' ; and so get a series of division marks 

 p', q,' r' , s', and t' on cA, dividing bd similarly at p' (such 

 that DP'=Ap'), q', r', s', and t'. - »- 



Now join FE, gg', hh', kk', ll', ikc, pp', qq', rr', Ac. 

 The lines thus drawn will envelop tangentially a quadrant 

 of an ellipse having cb and ca as semi-axes. 



If, having carefully made this construction in pencil, we 

 prick off corresponding division marks on the two diameters 



Fig- 2. 



of which f E and cf are the halves, and on the four sides of the 

 incloi-ing rectangle (bd and da being halves of two of tliese 

 sides), prick off division marks corresponding to those on 

 DY, . DX (ydx is half a square), we obtain the tangential 

 envelopes of the complete ellipse. The resulting figure has 

 a somewhat singular appearance, the ellipse seeming strangely 

 cut out from within the inclosing circle of which fde is a 

 fourth. 



The same figure may be formed, more quickly, by making 

 two direct tracings of fig. 2, and two inverted tracings 

 (these can be made by turning the figure with its fate 

 towards a glass sheet, and tracing from behind by n^eans of 

 the light received through the glass), and combining these 

 four figures info one. 



III. The Hyperbola. — There are sevei-al ways of obtain- 

 ing an hyperbola by means of a series of enveloping tangents. 

 Hereafter I shall introduce a prettier and more symmetrical 



method than that which I present here ; but this will serve 

 as a convenient companion to the methods used for the other 

 conic sections. 



Let c be tlie centre, ca a femi-axis, and CD, cji' the 

 asymptotes of the proposed hyperbola. Along nc take the 

 points E, F, G, nearing each other towards c. Let parallels 

 (not drawn in the figine) through d to d'e, d'f, and d'g cut 



Fig. 3. 



cd' produced in e', /', y' ; and make the divisions along C(/ 

 and eg' ahke, in the points e, f, and </, g' f', and e'. Then 

 join by straight lines go', JV, eE', dd', Ee', f/'', and g//'. 

 These lines will envelop the required hyperbola — that half 

 which lies within the space goj'. The other half can be 

 similarly obtained. 



THE SCHOOLGIRLS' PUZZLE. 



^ HE problem of arranging fifteen schoolgirls 

 so that they may walk out in sets of three, 

 seven days in succe.'sion, no two giils bein" 

 in the same set of three twice, has been 

 rather fully dealt with in Vol. I. (No. 9) 

 Vol. If. (No. .35), and Vol. III. (Nos. 73 and 

 79 — twenty-one schoolgirls to walk ten days 

 in sets of three). The general discussion cannot, I find, be pre- 

 sented in a manner suited to these columns, as considera- 

 tions of some complexity presently arise. But the followin" 

 points may be noted in regard to problems of this class. 



First, take such a problem as the pairing of an even 

 number of chefs. players as in tournaments, where each 

 player is to meet one opponent each day till he has met all, 

 the encounters on each day all beginning at the same time. 

 Or, if we must have schoolgirls in the puzzle, let an even 

 number of schoolgirls have to walk in pairs, day after day, 

 each having a different partner on each day until she has 

 walked with all the rest. 



This puzzle or problem can always be very easily soWed 

 as follows. 



