214 



KNOWLEDGE - 



[Jan. 6, 1882, 



togcDior, tlioy <lo not rrmnin so, preferring to ocparato into in- 

 niimcnible Kfo"!'"' '''''' f""iil'f» or tribes, and sometimos making 

 the lont; transit in companies of two, three, or five. While 

 remarkable for the jiowcr nutl speed of their flight, they become 

 fatigued in passing the sen, and will Hock in great numbers 

 npon tlie rigging of a ship passing their course for a rest. 

 Sometimes tlie birds are so utterly worn out with fatigue, that when 

 they have perched npon the side of a boat they are unable to take 

 again to the wing, and, if disturbed, can scarcely fly from one end 

 of the boat to the other. They have even been seen to settle upon 

 the surface of the waves, and to lio with outspread wings until 

 they were able to resume their journey. Guided by some won- 

 derful instinct, the swallow always Ends its way back to 

 the nest which it had made, or in which it had been reared, 

 as has frc(|uently been proved by afhxing certain marks to 

 individual birds and watching for their return. Sometimes it 

 happens that the house on which they hud built has been taken 

 down during tlieir "season abroad," and in that case they 

 exhibit a most pitiable distress, flying to and fro over the spot 

 in vain search after their familiar domiciles, and filling the air with 

 a mournful cry, announce to their friends that they have been dis- 

 possessed or cvict_Hi in the interest of local improvements. The 

 swallow is widely spread over various parts of the world, being 

 familiarly known tlironghout the whole of Eurojje, not excepting 

 Norway and Sweden, and the northern portions of the continent. — 

 Frank LesUe's Magazine. 



#ur iWatlKmatical Column. 



MATUEMATICAL QUERIES. 



[14] — Can you inform me — (1.) Whether the axis of any cone 

 passes through one of the foci of every ellipse formed by a section 

 of that cone ? (2.) Whether the two ellipses formed by the section 

 of the two cones having a common apex and a common axis, by the 

 same plane jiassing through both, at aiiy angle to the common 

 axis, are of similar eccentricity ? (3.) Whether or not the angle of 

 inclination of the ecliptic to the axis of rotation of the sun bears 

 the same relation to the eccentricity of the earth's orbit, as the 

 angle which the plane of any ellipse forms with the axis of its cone 

 bears to the eccentricity of such ellipse ? — No Mathematician. — 

 [None of these relations hold. The simplest way to determine the 

 foci is this : — Take a i)lane through axis of cone and at right angles 

 to the cutting plane. A circle inscribed in the triangle in which 

 the plane cuts cone and cutting plane will touch the axis of ellipse 

 in one focus. The escribed circle touches it in the other. — Ed.] 



"T. B." sends an ingenious solution of No. 7, p. 148 (Know- 

 ledge, No. 7), in which he claims (erroneously) that no proposition 

 beyond those in Euclid Book II. is employed. We have slightly 

 modified the construction in what follows, in order that the figure 

 may bo more conveniently sha))ed, but the solution is in effect the 

 same that " T. E." has kindly sent us. 



E M 



We have in right angled triangle BAC, AD perpendicular to hypo- 

 thcnnsc, DM, DS perpendicular to BA, AC ; and wo have to show 

 that angle /}J/C=anglo BNC. 



Kect. BM. MA = MD' (whereabouts in Books I. and II. is this 

 proved ? It might bo given as a corollary from II., 14, but not 

 witboot some proof bringing it within the range of Book III.). 

 Hence, adding MA' to each, we have — 



BA. ^M=.lD' = (simiIarly) AC. AN. 

 Complete rectangle .^NO/J, take j4/i = .4.U; draw h'L parallel to .•IB, 

 cutting BN in F ; and draw Kt'Q parallel to AN. Then rect. E0 = 

 rect. AL (K4 and FO being complementary) ="Iii4. >IA'=CV1. AN. 

 Hence Bh! must bo equal to ,iC; and EF'^MA. llenco triangle 

 Bi'J' is equal in all respects to triangle CAM. Thus angie ACM 

 (=alt. angle C3fi)) — angle i'l(>'=alt. angle BNO. Adding a right 

 angle to tlio equal angles CM D an(\ BNO, y/e have angle BMC" 

 angle BNC. — QHD. The jiroof is not so easy as cither of those we 

 gave, but it illustrates a useful method. — Ed. 



W. Ridd obtains a result, in examining the problem dealt 

 with in (pierj- 92, p. 115, slightly different from onrn. Wo 

 gave for the eastwardly defli-ction of a pmjectile lot fall from 



a height A, tl cos .\ where (is the time of the fall, Xtho latitude, 



and P the earth's rotation-period. He gets instead zZlL LL cos X 



Mr. Ridd overlooks the circumstance that the point below moves 



eastward at such a rate as to bo carried a distance 



2Trr( CDS X 



ward in time /, so that the actual eastwardly deflection is only the 

 difference of these,'or ~J[_ — ^ _ The result ia not slightly, but 



very, different from that wo gave, being more than — timcsas great, 



h 

 so that if h be 88 yards, W. Ridd's result would be greater than 

 mine in the same degreelthat the earth's radius, or about 3,9C0 miles, 

 exceeds 88 yards, or 3,960 x 20 times, or 79,200 times ! In fact, Mr. 

 Ridd's error is the converse of Tycho Brahe's, who, in a letter to 

 Ilothmann, asked, " how it was possible that a ball dropped from 

 the summit of a tower should always fall close to the foot of it, 

 since the tower must have moved a considerable distance towards 

 the east while the ball was falling ; if the height of the tower were 

 WO feet, the falling body should strike the ground IJ miles west- 

 ward from the foot of the tower, which is contrary to all observa- 

 tion." 



But, as a matter of fact, the result we gave is only correct when 

 we neglect the circumstance that during the fall the direction of 

 gravity on the falling body varies, so that — first, the direction of the 

 body's excess of eastwardly motion over the eastwardly motion of 

 the point vertically below the point of suspension, is not always at 

 right angles to the moving vertical, and, secondly, gravity acts 

 during the fall to partly diminish this part of the motion. These 

 may seem very unimportant matters, but, as a matter of fact, when 

 they are taken into account, the calculated eastwardly deflection is 



27r?itcosX 47r/jfcosX 

 found to be diminished from p to ^p . 



We leave it as an exercise to the student to 

 obtain this result by analytical methods. (If any 

 difficulty should be found, we shall be glad to 

 give the solution.) The following geometrical 

 method will be readily understood by a larger 

 number : — 



Let .4 be the point of suspension, B the point 

 vertically below it, C the earth's centre, BEF the 

 earth's surface along a great, circle tlirough £.4, 

 and touching the latitude-parallel (or small circle) 

 through i), so that BF may be regarded as part 

 of this latitude-parallel. 'The body falling from 

 .4, with such eastwardly motion as belongs to the 

 point of suspension ,4, travels in an elongated 

 ellipse, AFA', having L', the earth's centre, as a 

 focus, and reaches the ground at F, the arc APF 

 being appreciably jiarabolic. Suppose that while 

 this descent is taking place the point of suspen- 

 sion, A, is carried by the earth's rotation to D. 

 and join DC and FC, DC cutting BF in E, and 

 Al'F in P. Also let arc AD produced meet OF 

 produced in (t. 



Then, since the point of suspension A, and the 

 falling body when just leaving .4, are sweeping out 

 equal areas around C, and continue to sweep out 

 areas uniformly during their motion (the former 

 because of the uniform rotation of the earth, 

 the latter by Kepler's second law), it follows 

 that 



Area ^ CO = area APFC. 

 Whence, taking away from each the area APC 



Area ^PD= area CPF. 

 Whence approximately (since PE and EF are each verj- small, 

 compared with .4U, BF, &c.) 



Area .41) = area CEP, 

 1 1 



or, approximately, s^^- BE — 5Ef.BC 



2.iB.BE. 

 Whence Ef =„ 



3 BC 



or, since .4B=/i, and B£ = 2»-i- cos X 

 2 h 

 °3 

 tI,I 

 ■JP 



(f) 



the easterly deflection i'E=^ • -• 2aT cos X./^— J 



-cos X 



