Feb. i.'4, 1882.J 



KNOWLEDGE 



369 



side ; for the hyperbola, the second sphere touches the cone on the 

 oilier side and the plane of section on the same side. 



The reader will tind no difficulty in extending the proof to the 

 liriis IfP, PM', drawn from a point P on the curve to the other 

 torus U and perpendicular to the other directrix X'M'. The con- 

 struction is given for each case. 



Note also that the relations HP+SP = AA' for the ellipse, and 

 HP-SP = AA' for the hyperbola follow at once. 



For, rotating the curves back to the foreshortened view, we have 

 SP foreshortened into tang. XS = tang. X/i = aK 

 HP foreshortened into tang. NH = tang. Nii'=bK 

 Wherefore HP + SP = ab in case of ellipse (Fig. 1) 

 and UP - SP = ab in case of hyperbola (Fig. 3). 



— EriTOR. 



The geometrical student will find a good deal more that is worth 

 studying in the relations here indicated. We have added several 

 lines (latus rectum, minor axis, &c.) to the figures for this purpose. 



MATHEMATICAL QUERIES. 



[37] — Given any two lines meeting in a point, and some point 

 out of the lines ; required to draw from this point to the str. lines 

 two equal str. lines which include a given angle. — Amicus Mathk- 



MiTIC.E. 



[38]— Rolling Disc. — Given the radius, weight, velocity, and 

 angle of inclination sideways from the vertical of a circular disc 

 rolling freely on a level piace; find the radius of its track. — 

 F. W. F. 



[31]— a'' + 4r3 = 27 



put i' = 3!/, then 3!(' + 4i/'' = l 



and y = -, then 3 + 4: = ;* 



Make a perfect square on each side — 



(a) :* + 2p2' + p' = 2p:^ + 4c+i)' + 3 

 The right hand side will be a perfect square if — 

 i=2pip' + 3) 

 i.e., a p^ + 3p = 2 



puti) = i' — -, then p^ = V — 3p 



V V 



taking the upper sign v' = v'2 + 1 

 and — =v'2-l 



Let p have this value, then, from equation (o) — 



A quadratic with two roots corresponding to each sign, thus giving 

 four values for ;, and, therefore, four values for x (ic= _ , as might 

 have been anticipated. — W. G. 



[Equation, p. 328, No. 15]. — " W. B." points out that in our 

 solution of equations 



, 39 14 , J 42 13 

 x- = — — — and i/= _— — 

 y X X 1/ _ 



after getting (x + i/)' = 216 by addition, we might have got by 

 subtraction (x — y)' — 8. It is obviously the simpler course. — Ed. 



22!). — " Yarletonian," F. J. Butt, and others solve this equation ; 

 it needs only transposing, squaring, and simplifying, then squaring 

 again and simplifying. 



[Mr. McGowan's solution to 25, p. 307, to hand, correcting obvious 



blunders (20 for 120, and £,- for £-). It shall appear in our next. 



5 6 

 —Ed.] , 



Messrs. J, & A. CnUBCHiiiL have recently published two interesting 

 tables ; one, showing the average weights of the human body and 

 brain, and of several of the internal organs at eighteen jicriods of 

 life in both sexes ; the other sho«ing the same (at decennial periods 

 of life) in the insane, the forms of insanity being specified. 



©ur Cfjrsss Column. 



Xo. 21. 

 By I. G. 



No. 22. 

 By W. Thurman. 



[AYliite to play and mate iutwo moves. White to play and iimtf in two moves. 



SOLUTIONS. 

 Problem No. 14, p. 282. 



1. B. to R.4. 1. K. takes Kt. 



2. B. to Kt.3. mate. 



If P. to K. 5, then Q. takes P. mate. If P. to Q.3 : Q.Kt. to B.7. 

 mate. If P. to B.3.' K.Kt. to B.7. mate. Finally, if P. to B.4. then 

 Q. to Kt.8. mate. 



Problem Xo. 15, p. 282. 



1. Q. toQ.6. 1. K. to K.6. 



2. R. to Kt.3.ch. 2. B. takes K. mate. 

 If K. to B.6. then R. to B.3.ch. 



Problem Xo. 16, p. 308. 



1. Kt. takes P. 1. K. to K.5, or a, b, c. 



2. Q. to K.G.ch. 2. K. to Q.6, or B.C. 

 Kt. mates accordingly either on Q.Kt.4. or K.R.4. 



(•) If 1. K. to Kt.7., 2. Q. to K.2.ch., 2. K. to K.6., 3. Kt. to B.4. 

 mate, or 2. K. to Kt.8., 3. Q. to B.2. mate. 



('') If 1. K. to Kt.5., 2. Q. to Kt.6.ch.,2. K. to B.6., 3. Kt. to Q.4. 

 mate, or 2. K. to R.6., 3. Q. to Kt.3. mate. 



(■) If 1. B. to Kt.7., 2. Q. to Q.B.4. anything, 3. Kt. to E.4. mate. 



AXSWERS to CORRESPONDENTS. 



*,* Please address Chess-Editor. 



Edward Sargent.— Nos. 18 and 19. If 1. Q. takes B.P., then 1. Q. 

 to Q.R.sq. 



H. S. Standen. — Solution of No. 15 correct. 

 C. H. F. — Solution of No. 17 correct. 

 H. .\. L. S.— Solution of No. 17 correct. 

 J. P.— Remove Pawn on Black Q.R.2. 

 F. H. Jones. — Solutions correct. 



Received offers to play by correspondence from- 



M. J. Harding 

 H. C. Angell 



E. A. Dillon 



F. H. Jones 



A. C. Skinner 

 J. N. Siclebotham 

 Edw. P. Westlake 

 D. Cudmore. 



We have paired them in tiie order named above. 



It is necessary the first players should play White. Two games 

 mav be carried on simultaneously, each jilayer having the move. 

 Answers should be sent nest day after receipt of move, at latest. 

 To avoid mistakes, the last move should always be repeated. For 



P. to B.3. P. to. K.5. 

 example : — 12. „ t. o 13. In case of any misun- 

 derstanding arising, players may refer to us. 



A SOCIETY to be called the North Middlesex Natural Histor\' 

 Association has recently been established. Address, 26, Ingleby- 

 road, Grove-road, Holloway, N. Its objects are the formation of a 

 X'atural Historj' Museum and Library of reference and circulation; 

 also the diffusion of natural history knowledge by means of lectures, 

 papers, ic, and (in the summer) field excursions. 



