May L"), 1885.] 



♦ KNOWLEDGE ♦ 



423 



To give a single illustration : y Draconis is in the zenith of London 

 at its upper culuiication, and never sets there ; continuing to cirolo 

 round and round the pole. It never approaches nearer than 13' to the 

 horizon. Do let me beg of you not to continue to send nu> quires of 

 stuff that is really not worthy of e.\amiiiation by any thinking num. 

 — O. B. I have not the most distant notion who the I'rofessor cinoted 

 by Macniillan is — or was. There is an uniount of turgidity in the 

 quoted specimen of his diction, though, leading irresistibly to the 

 inference tl>at bis "eminence" may possibly be of a debatoablo 

 character. The almost equally sensible lines to which you subse- 

 quently refer as " nonsense," were written by Poote, the actor, to 

 test the memory of Macklin, who asserted that he could repent 

 anythititi by rote after once hearing it. 



(Piir iHatGrmattral Column. 



MATHEMATICS OF METEORIC ASTKONOMY. 

 By Richard A. PiiocToa 



{Continued from p. 4U3.) 



LET us next take the case of one of the giant planets, for 

 example the planet Jupiter. Here we may conveniently use 

 equations we have already obtained for the sun, substituting the 

 proper values for Jupiter. Thus, the mass of Jupiter is 1-lOlSth 

 of the mass of the sun, so that fi for Jupiter 



For a body starting from rest at an iuGnite distance, and reaching 

 Jupiter under his sole attraction, we obtain the equation (as 

 before) — 



X 



And taking 86,000 miles for the mean diameter of Jupiter, wo 

 obtain for the velocity with which a bcdy approaching Jupiter 

 from without would reach the planet's surface 

 2x29,870000^597.10^ 

 43,000 43 



or i'=37-J miles per second. 



Let us next see what would be the actual increment of velocity 

 produced by the action of Jupiter, supposed to be at rest, on a body 

 starting at rest from an infinite distance towards the planet as at 

 J (Fig. 3) in the direction A J S, S being the suu. 



A PQ J 



I r • • S 



Fig. 3. 



Put JS = R; PS = x = Jupiter's distance from the sun = 

 481,780,000 miles; PQ = d ,t ; then, we get (as when dealing with 

 the earth) the equation 



\dt) ~ X ■*■ (x-W) "^ ^ 



where \>. = 31,298,800,000 and /»' = 29,870,000. 



When the body starts from rest at an infinite distance, we have 

 = 0. Wherefore, in this case, 



,_2n^ 2m' 



a; - K 



Now, put (i-R) = Jnpiter'sradius = 43,000; and 1=481,780,000 <- 

 43,000 = 481,823,000. Then we have 



2 X 3 1,298,800 ,000 2 x 20,870 ,000 



^"^ 481,823,000 ■*" 43,000 (") 



= 129-9 + 1389-3 = 1519 2 

 y = 39 miles per second. 



In other words, the velocity with which a body travelling to 

 Jupiter and the sun, supposed to be both at rest and along its iim; 

 of approach, would reach Jupiter's surface under the combiiieil 

 influence of the attractions of both these orbs, would bo but 

 If miles per second greater than the velocity with which a body 

 would reach Jupiter under his own attraction alone, the approach in 

 both cases being made from rest at an infinite distance. 



Observe, again, that the number 129'9 represents i-' for a body 

 approaching the sun (from rest at an infinite distance), when at 

 Jupiter's distance from him. The velocity of a meteor when at 

 this distance from the sun, if the meteor has simply been drawn 

 from out of interstellar space, is therefore >/129'9, or 11* miles per 

 second. 



Wo notice that under the conditions hero specified .Inpiteradds 

 a velocity of 27J miles per second (3!) — llj) to the velocity with 

 which, were Jupiter away, a body would roach Jupiter's distance 

 from the sun under the sun's attraction only. Wo notice, also, 

 that a body ejected from Jupiter placed as at J (l''ig. 3), so as to 

 start from Jupiter's surface towards .'V with a velocity of 30 miles 

 per second, would pass away into interstellar space, never return- 

 ing to the planet. 



And just hero we may for our amusement, ask ourselves what 

 would happen if a body set out from Ju|)iter's surface on the side 

 towards S (Fig. 3) with this same velocity of 39 miles per second. 

 J PQ R 



~ • 



Fig. 4. 



In Fig. 4, let J S = 181 ,780,000 ; V S = x ; V q = d.r. Then, 

 since now Jnpiter's action t(mds to retard tno motion of tho body 

 on its way towards S, the equation of motion is 

 d'x _ _ ;« , ^' 

 J?" ;>:' (R-x)-' 

 whence, as in the other cases, — 



'dx\\ ._, 2,i ~ ■ 



(^y 





R- 



--fC. 



= 39, or "- = 1519-2; so 



1519-2 = 



+ C 



Now when i = R-43,000 = 481,737,000, 

 that 



31,298,800,000 2 « 29,870,000 



481,737,000 43,000 



= 120 9-1389-3 + 

 or = 2778-6 



Now substitute this value of C in the above e(iuation, putting 

 J = sun's radius = 430,000, and we have 



2 X 31.298,800,000 2 x 29,870,000 , .,,,,„ „ 



V- = — +-/7ot> 



430,000 481,350,000 



= 145576-0-1 + 2778-6 

 = 148,354-5 

 or V = 385 miles per second. 



Thus the velocity at the sun's surface is increased by only 3i 

 miles per second (385 — 381i), in the supposed case, viz., where 

 the body starts from Jupiter towards the sun with the same 

 velocity which it had acquired when it reached tho surface of 

 Jupiter, coming from rest at an infinite distance under tlie com- 

 bined influence of Jupiter and the sun. We can easily see why, in 

 this caiie, Jupiter imparts a greater velocity to tho approaching 

 body, than be withdraws as the body is receding. For the body's 

 sun-imparted velocity is less before Jupiter is reached than after 

 Jupiter is passed, and the less the velocity of the body the longer 

 time is it under Jupiter's influence, and the greater therefore the 

 resulting change of velocity. Thus the increment of velocity is 

 somewhat greater than the decrement. 



Beit now noticed that in every casein which a meteorite coming in 

 from outer space passes close to Jupiter, there will be an acceleration 

 of the meteorite followed by a retardation, or a retardation followed 

 by an acceleration, and only the balance of eUeot is to be regarded 

 as the permanent acceleration or retardation of the meteorite. The 

 above investigation shows that the total increment of velocity 

 (27^ miles per second) in a ease manifestly giving a result beyond 

 the maximum possible increment, is much less than the total 

 velocity which Jupiter can impart in a body approaching him from 

 an infinite distance under his own attraction only. And we see 

 further that the velocity subsequently withdrawn is almost as great 

 as the velocity thus add>-d. It will be obvious, without independent 

 investigation, that in the caso of a meteorite which does not come 

 within a considerable distance of Jupiter's surface, tho velocity 

 added or abstracted by Jupiter will be much ]e»s, as will also 

 be the velocity abstracted or added, and consequently tho balance 

 of velocity whether added or abstracted. 



Thus, suppose we want to determine the velocity of a body 

 approaching, as in the case illustrated by Fig. 3, when at a distance 

 of 4 radii of Jupiter, from Jupiter's surface. We get, obviously, 

 instead of equation a above, 



V- = 120 9 + i (13S0-3) = 477-2 

 or V = 21-8 



this is an increment of 104 miles per second (218 — 11-4) on the 

 velocity which the sun's attraction alone would have imparted. 

 The velocity subsequently abstracted would bo nearly as great ; 

 arid the balance of increase, therefore, relatively small. So, also, 

 the balance of decrease, even in the case of so near an approach to 

 Jupiter as this, would b") small. And the chance of a meteorite 

 coming so near as this to Jupiter would be almost inconceivably 

 small ; while the change of velocity unfler tho actual conditions 

 would be in reality much less than in the imaginary case here dealt 



