18 



• KNOWLEDGE ♦ 



[Jdly 4, 1884. 



annnlus or ring of light that surrounds her orb, which is proof 

 positive and complete that she is enveloped in an atmosphere 

 similar to our own. The same arguments were used long ago by 

 Dr. Dick and others to prove that Jupiter and other primaries are 

 destitute of an atmosphere, while the very fact that they are 

 visible, is evidence that they are endowed with atmospheres like 

 that of our own globe, for surely no one is now antiquated enough 

 to believe that light exists throughout space but only in the atmo- 

 sphere of the planets. The non-existence of a second satellite to 

 the earth is therefore not proved by reason of its invisibility. Its 

 position can only be ascertained by noticing the sudden quenching 

 of stars, and I trust that astronomers both in America and Europe 

 will aid in obtaining the magnitude and motions of this lonely 

 wanderer in the sideral heavens. E. Stone Wiggins. 



Ottawa, June 3, 18S4. 



To all which nonsense the Neiv Tori Tribune is at the pains 

 somewhat gravely to reply. The closing words of the leader de- 

 voted to this precious rubbish are neat however : — " Unfortu- 

 nately," says the Tribune, "there is one view of Mr. Wiggins's 

 discovery which he lias failed to take, to wit, the view that it may 

 not be at all necessary. For between accepting a dark moon (save 

 as an exercise of pure faith) and believing that the meteorological 

 theories of Mr. Wiggins are nonsensical, the great majority of man- 

 kind will, we fear, be very apt to find the second conclnsion the 

 easier and simpler of the two." 



There is a series of illustrated notes on the Pons-Brooks Comet 

 of 1883, by H. C. Wilson, in the Sidereal ilefseitrfer for June, which 

 students of cometary physics will read with interest. 



Manganese in Animals and Plants. — Becent researches by M. 

 Maumene have, says Engineering, shown that the metal manganese 

 exists in wheat, rice, and a great variety of vegetables. Wheat 

 contains from .jjj'^^ to tsj^^jj of its weight of the metal, which 

 exists chiefly as a salt of an organic acid. It is also found in 

 potatoes, beetroot, carrots, beans, peas, asparagus, apples, grapes, 

 and so on. The leaves of the young vine are very rich in it ; so 

 are the stones of apricots. The proportion in cacao is very great, 

 as it is in coffee, tobacco, and especially in tea. In the 50 

 grammes of ashes left bj' a kilogramme of tea, there was found 

 5 grammes of metallic manganese. There are vegetables, how- 

 ever, in which no manganese can be found, as, for example, 

 oranges, lemons, onions, tfcc. Many medicinal plants contain it, as 

 for example, cinchona, white mustard, and the lichen (Roccella 

 tinctoria). Animal blood does not always contain it, but it is 

 found in milk, bones, and even hair. M. Maumene regards its 

 presence in the human body as an accident, and not of vital im- 

 portance. He also suggests that doctors should cease to employ 

 manganese as a succedaneum with iron, for while the latter is 

 useful to the blood, the former is an intruder which is only tolerated 

 in small traces, and rejected in larger quantities. Tea, coffee, and 

 other vegetables require abundance of manganese in the soil for 

 their proper cultivation, and the absence of it may account for the 

 failure of many plantations. 



In connection with the series of lectures now and for some time 

 past in course of delivery by some of our best -known scientific men, 

 at the Royal Victoria Coffee Hall, a lecture was given on Tuesday 

 week by Mr. Arthur Nicols, F.G.S., F.R.G.S., on " The Dog as the 

 Friend of Man," illustrated by numerous large coloured pictures 

 of the heads of the principal breeds, prepared specially from draw- 

 ings by distinguished artists. The lecturer treated his subject from 

 the point of view of the lover of dogs, rather than of the dog- 

 fancier. He first gave a sketch of the origin of domestic dogs, 

 which all competent naturalists are now agreed in considering as 

 having been derived from some three or four wild species ; and pro- 

 ceeded to consider in detail the senses — sight, hearing, and smell — 

 by instances mainly derived from his own experiences at home and 

 abroad, giving many illustrative examples and anecdotes of the 

 utility of dogs. In more than one instance the lecturer showed how 

 his life had been saved by the viligance of these faithful animals. 

 He next described the characteristics of the principal breeds, com- 

 mented on dog-shows, canine madness, &c., and gave instances in 

 evidence of the intellect and moral character of dogs, concluding 

 with remarks on the influence which association with the dog has 

 exerted on man himself. The lecture was listened to throughout 

 with great attention by a very considerable proportion of the 

 audience, the conduct of the occupants of the gallery, however, 

 leaving much to be desired. A hand-bill was previously distributed 

 in the hall of notes, drawn up by Mr. Nicols, on " Mad Dogs : How 

 to Know Them, and What to Do," with the object of diffusing 

 useful information on this important subject. 



0av iHatl)tmattraI Column. 



EAST LESSONS IN CO-ORDINATE GEOMETRY. 



By Richard A. Peoctoe. 



(Continued from p. 467.) 



PoLAB Co-OEDINATES. 



45. Pbof. — To find the polar equation to a straight line in terms 

 of the angle at tchich it is inclined to the initial line, and the intercept 

 on the initial line. 



Let A B be a line meeting 

 the initial line OX in A. 

 LetOA = a, and /BAX = a. 

 Take P any point on A B, and 

 let the co-ordinates of P be r, 

 9. Join P, then 



sin OAP 



P = OA 



sin OP A 



,, . . sin a 



that IS r = a ^ 



sin (a-e) 



the required equation, which 

 may be written in the form 

 r sin (9 — a) -l-a sin a = (1.) 

 We might have obtained (1) 

 from the equation to A B in 

 rectangtilar co-ordinates (O X 

 the axis of X). For draw O K 



perpendicular to X to meet A B in K, then O K = a tan a. Thus 



the equation to B K is 



— —r^- =1 

 a a tan a 



that is, since x = r cos 6, and y = r sin 



T cos r sin cos a 



r sin (9- 



-1 = 

 a am a 



-a) + a sin a = 



(1) 



the 



or 



as before. 



We can obtain the polar equation iu a more convenient form by 

 determining the line in a different manner. 



47. Pbop. — To find the polar equation to a line in terms of theper- 

 pend icular on the line, and the an/jle at which this perpendicular is 

 inclined to the initial line. 



Let A B be a straight 

 line. Draw Q perpendi- 

 cular to A P, and suppose 

 OQ=p and ZQ0A = a; 

 let r. 9 be the co-ordinates 

 of any point P in A B ; 

 then 



0PcosP0Q = OQ 

 that is 



r cos (9 — a) =p 

 the required equation. 



47. If in (1) a = 0, 

 equation becomes 



rsin (0-a)=O 

 ie., sin (6 — a) = 



Hence S = a or else 9 — a = Tr; that is 9 = a or a + jr; and it is 

 obvious that either form expresses the same line. Hence the 

 equation to a line through the pole inclined at an angle a to the 

 initial line is 



= a. 



4S. And, vice i-ersS, an equation of the form 6=a constant, repre- 

 sents a straight line through the pole ; for it is clear that such a 

 line is the only curve for every point of which 9 is constant. 



49. The polar equation of the straight line is of the form 

 Ar cos0 + Br sin0-i-C=O (1) 



and we might easily prove, conversely, by an independent process 

 that an equation of this form always represents a straight line. 

 This is not necessary, however, since transforming to rectangular 

 co-ordinates making the pole the origin and the initial line the axis- 

 of X, (1) becomes 



AI-^Bl.;-^C=0 

 an equation which, as we have already seen, represents a straight 

 line whose intercept on the axis of ^r (that is, on the initial line of 



C 



our polar equation) is — 5, and which is inclined to the same axis or 



B 



initial line at an angle whose tangent is — t- 



