April 17, 1SS5.] 



. KNOWLEDGE ♦ 



335 



these columns, it might — in fact wo<iUl — drift auvwhorc. Uocause, 

 a^ I have said, I must utterly refuse to listen to one side only ; and 

 if \ve are to have Atheism, why not the Athauasinn Creed, too P — 

 Alfred Cappek. Thanks to Easter holiday-making (not my own, 

 by the way), your card of imntation reached me too late to be 

 utilised. — F. W. Hiulek. Keceived with thanks. — U.vilk John. 

 All that we can detinitely say is that light, heat, electricity, &c., 

 are theoretically explicable on the assumption of nndulntions or 

 vibrations in something. What that "something" is, though, we 

 are in the most profound ignorance. It behaves, in some sort, as 

 though it were solid 1 Vour idea that it resembles in any way an 

 excessively attenuated gas is quite erroneous, and we have, hence, 

 not the slightest reason to believe that it, as it were, brushes off any 

 part of the bounding surface of our atmosphere. — Some Anonvmois 

 CoBBESPO.VDEXT s?nds me a cutting from a periodical cf an answer 

 to " John Jones," which is — not to put too fine a point on it — sheer 

 nonsense. — Thomas Black. De ■justibiis non cft dispiitandum. The 

 only explanation which occurs to me is that the present tense 

 imparts more vividness. — Thomas Ayers. Presumably you mean 

 p. 327, and not 237 as yon write. Well, you were correct in your 

 surmise as to the author of that reply. I meant your letters of the 

 last six months. You are obviously one of the noble army of 

 the Rreat unappreciated. — Lauy Anne Dindas. I regret that I 

 am ignorant of the address at which Dr. Carter Moffatt's ammonia- 

 phone is sold. Does the need for reticence with reference to the 

 Sea-serpent still exist ? Your experience would probably prove most 

 interesting. — Geo. St. Clair. Y'ou should consult the Procecdiyigs 

 of the Koyal Society for any and all Dr. Huggius's spectroscopic 

 observations of the heavenly bodies. — W. R. K. Mix your lamp- 

 black with gold-size (not common size) by the aid of a spatula, and 

 mind that you only put just enough gold size to make it stick 

 together. Add about twice the volume of turpentine and paint on 

 with a camel-hair brush. — Reform. I know nothing of where 

 Morrell's aah-closet is procurable. Perhaps its inventor will 

 advertise his address. The word between the two you specify 

 is " Panel." — Miss Rosalixd Cloctte sends, for Miaa Ballin's 

 edification, a story of a fox-terrier, " Grip," who, having followed 

 a cat into a churchyard which he was forbidden to enter, and, 

 being caught jiagi-antc delicto, went, unbidden, and fetched a 

 stick for his mistress to beat him with ! — J. Murray. None are so 

 blind as those who won't see. If the earth were approaching the 

 ann at the rate yon imagine, do you not perceive that the year must 

 shorten very notably and perceptibly indeed, after a comparatively 

 short period ? As for Dr. Hind's calculations, tbey are one and all 

 based on the assumption that the distance of the earth from the 

 sun is (practically) invariable. Were there a scintilla of truth in 

 your "P. D." stuff, every one of his results would be hopelessly 

 wrong. — Thos. Fcrxell. You write as though you were the 

 solitary correspondent of this journal ! — David Chadwick. I cannot 

 reprint letters which have previously appeared in contemporary 

 columns. — C. E. Xotts asks Mr. Browning to inform him " what is 

 the best form of tandem tricycle for a gentleman and lady ? " I 

 will certainly not give yon Mr. Bro^vning's private address. It is a 

 liberty which I would not take with any contributor to these 

 columns. — Dr. E. Wooton. Shall have immediate attention. The 

 pressure on our space is very heavy indeed just at present. — F. 

 W. H. Hylo-Idealism has already been sufficiently expounded here. 

 It is not difficult to make a precis even of your own epitome of it. 

 Here it is. The universe is the sum of a man's sensations : destroy 

 the man and you annihilate the universe. — Commentator accuses 

 the conductor of this journal of inconsistency in having, a few 

 months ago, spoken of Mars and Venus as habitable, if 

 not even inhabited; and now contending that the earth 

 is the only body of which habitabilit}^ can be even plausibly 

 predicated. He regards this as even more " dismal astro- 

 nomy" than that of Mr. Mattieu Williams, and contends that 

 evolution teaches ns that " Form and life are possible under all 

 sorts and conditions of being." "An Earnest Thinker" has never 

 sent for your book. Thanks for the photograph. — An Anoxytiocs 

 CoEEESPOXDENT Ecnds me the Ulverstone Mirror, with an article on 

 " The Salvationists and Faith-healing," reprinted from the Man- 

 cliester Courier. As a trenchant exposure of a very pitiful humbug, 

 this is well worth reading. The paper also contains an account of 

 a slate-writing aiance given by Mr. T. W. Gowland, of Ulverstone, to 

 explain the modus operandi of */iai humbug too. From the published 

 account, Mr. G. seems to have been very fairly successful, though 

 perhaps not to have quite equalled Mr. Eglinton in the neatness 

 with which he performed all his tricks. But then Mr. E. has, of 

 course, had such enormous practice. Finally, the same paper 

 contains a categorial denial by Sir Henry Ponsonby that the Royal 

 Family are spiritualists j and a flat contradiction by the Hon. A. 

 Yorke, the Equerry to the late Duke of Albany, of the industriously- 

 propagated lie that the Duke ever held a stance with Mr. Eglinton, 

 or presented him with a slate. 



([Par iHatOcmatiral Column* 



MATHEMATICS OF METEORIC ASTRONOMY. 

 By Richakd A. Puoctor. 



AT the very foundation of all problems in nietoorio astronomy 

 lies the inquiry into the velocities wliieli can bo iinjjarted to 

 such bodies, as thoy ajiproach the neighbourhood of a sun or planet 

 under various conditions, — that is, when travelling with varying 

 velocities and in different directions. Some of these problems are 

 too recondite for these columns, but the simpler relations may be 

 dealt with in such sort that all who are familiar with the elements 

 of the differential calculus can understand them. 



Suppose that a body of mass 1 (i.e., the unit of mass) is moving 

 along the straight lino \ 1' S, Fig. 1, towards S, an attracting mass, 



PQ 



-«S 



Fig. 1. 



and that we call the distance P S, .r, and that when the body is at 

 P the time elapsed from some fixed epoch is t. Tlion if in a 

 minute increment d t of the time, the body passes to Q, P Q being 

 a minute decrement d x of the distance from S, we see that the 

 velocity along P Q, appreciably uniform, because of the minuteness 

 of both the time and space, is represented as usual by the space 

 divided by the time, — which means, of course, the number repre- 

 senting the space P Q divided by the number representing the time, 

 in whatever units of space and time we choose to adopt. Thus thou 

 calling the velocity r, wo have 



dt 

 We know also that whatever be the force (call it /) acting on the 

 body at P towards S, this force is uniform through the small time 

 t, and if Jv be the slight increment of velocity at the end of the 

 time dt, this increment may he regarded as added uniformly in this 

 time. Thus, just as in dealing with the uniforni action of gravity, 

 g, on a falling body, we say i' = gt, so we say for our particle 

 moving under a uniform force, /, during time, (, 

 dv = f . dt 



dv d fdx\ d'x 

 ' ^ ^di ^ di\di) ^di- 



Now, if S is a body attracting according to the law of nature, 

 that is, in degree proportioned directly to the mass and inversely 

 to the square of the distance, then the force acting at P, is propor- 

 tional to the mass of S -=- i-, and is negative, because the force 

 tends to diminish x. Thus, then, we write 

 d ,1! 11 



dt' X- 



(i) 



Where ft is a constant to be determined in each case from known 

 relations : it is in reality the number representing the velocity 

 generated by S in a unit of time on a body at a unit of distance (S 

 being supposed to remain constantly at that unit of distance from 

 the body throughout the unit of time), or better, the velocity gene- 

 rated in a unit of time by a constant force equal to that exerted by 

 S at a unit of distance. 



^dx 



To integrate (i), multiply both sides by (because in that 



dt 



way both sides can be integrated at once). Then 



„ dx d^x 2fi dx 



dt dt' 



dt 



integrating, (— ) = ^ + C = v- 

 \ dt/ X 



Suppose, now, that at some known distance r from S, the velocity 

 of the body is known to be u. Then 



„= = ^ 



+ C 



2f 



and 



Ki-J) 



(ii) 



This determines v whenever the velocity at some given distance is 

 known, if fi is known. Or if the body is approaching under the 

 sole attraction of the mass S, and the velocity at a given distance 

 is known, then /i can be determined, and thence the velocity at any 

 distance from S. 



As an example, take the sun. We know that our earth travel- 

 ling at a mean distance of 92,000,000 miles, and making one circuit 

 of herorbit in 3Go25 days, travels at a mean rate of nearly 18i 



