62 



♦ KNOWLEDGE ♦ 



[Jdly is, 1884. 



the truth of them to Rosah'e Vansittart and " An Old Draughts- 

 man" privately, ehouldjthat lady and gentleman desii'e it. — An Old 

 Draughtsman. Scarcely relevant to the general subject. — John 

 Bacon. With the excisions rendered imperative by the wholly 

 needless length of your letter, it has been marked for insertion. 

 Thanks for your offer of an article ; but original matter is provided 

 for some time to come. Besides, you must pardon me for saying 

 that you have yet to learn the art of writing briefly and to the 

 point only. — Jos. Lucas and Son. I strongly urge you not to 

 pait with anything to the person yon name, who is certainly an 

 impostor. No member of cur staff would be suffered (or, in fact, 

 would condescend) to accept goods, that favourable notices of them 

 might appear in the pages of Knowledge. — E. B. G. McDougall. 

 Thanks, but none such of any importance reach me. — W. J. 0. 

 suggests that the experiment should be tried of artificially extend- 

 ing the wings of a dead bird and attempting to fly it like a boy's 

 kite. — Maude Le Baille describes, at wholly inordinate lengh, how 

 she succeeded in teaching a little girl to read who had previously defied 

 the efforts of three or four governesses. Briefly she did so by first 

 teaching her to write. The child was allowed to draw on her slate, 

 as well as form letters and figures, the letters being later on com- 

 bined into monosyll.abic words. At this stage, to her own surprise, 

 she found herself capable of reading a page of " Mamma's Lessons." 

 No one but an editor would believe that this is told on nine pages and 

 a half (!) of exercise paper. — Willia.m Singer. As is so perpetually 

 the case with correspondents, tlie length of your letter forbids its 

 appearance. — Jas. Stanley Little intimates that the reviewer of his 

 book (on p. ]2) was in error in supposing him (Mr. L.) to be an 

 artist. He has never attempted to produce even the simplest 

 drawing since he left school. — St. E. There are but few works on 

 "animal" — as contradistinguished from human — physiology. 

 Probably Shea's " Manual of Animal Physiology," published by 

 Churchills, would be as good a book as you could obtain. — Rev. S. 

 B. Handley, and many other correspondents. As stated in my 

 reply to Messrs. Morris and Mackay above, I have entirely ceased 

 lecturing. Should I ever resume it, due notice will be given. — 

 G. C. I should be very sorry to sleep myself, and assuredly would 

 not suffer a child of mine to sleep, in a bedstead placed in the 

 position indicated in your diagram during a heavy thunderstorm. — 

 P. MacLeod Yearslev sends an anecdote of a dog which, finding a 

 piece of meat larger than it coidd eat, went to the very top of the 

 house to fetch a lady down to cut it up. — Wm. Wil.son, M.A., LL.D. 

 Your difficulty seems to arise from the omission by Colenso of the 

 word " whole " before number. Other writers on arithmetic add 

 this word. Of course, in one sense any number, whole or frac- 

 tional, which divides two or more whole numbers exactly, is a 

 common measure of them, and would be i-ightly so described. — The 

 Secretary of the National Smoke Abatement Institution. I 

 regret that absence from London prevented my attendance. — 

 E. C. E. Atmospheric electricity presents to a great extent still an 

 unsolved problem. It is, however, safe to assume that the electric 

 force, which would otherwise cause a dispersion of the cloud 

 particles, is feebler than the other forms of force which cause and 

 maintain a condensation of the originally charged attenuated 

 vapour particles. When the cloud charge attains a high potential, 

 it doubtless experiences a strong tendency to approach the oppo- 

 sitely-charged earth surface bodily. The intervention of the air 

 prevents this, and acting with the electrical attractions, the cloud 

 is broken up. Probably the attraction exerted by the opposite 

 charge on the earth prevents the breaking of the clouds in those 

 higher regions where, under ordinary circumstances, the discharge 

 of rain would commence, and eventually the drops assume larger 

 dimensions. — R. G. T. Madame Blavatzky's trick of causing a bell 

 to sound in the air may be bought at Hamley's, the Noah's Ark, 

 Holborn ; Bland's, New Oxford-street, or at any good shop where 

 conjuring apparatus is sold ; under the title " Is your Watch a Re- 

 peater ? " — Norwich. Many thanks for your kind and friendly 

 letter. Gossip has been rather crowded out for a week or two. 

 — Thos. Maclean. Believe me, I was not indebted to your sug- 

 gestion for the idea of the gyroscope. It has been a familiar one to 

 mo long enough. Alex. Gustafson sends me a little pamphlet on 

 the " Medicine Stamp Tax," which well illustrates the evils arising 

 from the quasi- Government sanction given to all sorts of quack 

 remedies. 



In reply to numerous letters and communications 

 addressed to tlie office of Knowledge, »'/.'( Editor hegs to 

 aiinou7ice that he has now concluded his Lecturiuj Tour, 

 and has, in fact, definitely ceased, to lecture altogether, 

 iiliould he {which is very douhtfid) at any future time 

 resume his lectures on Astronomy, due and ample notice 

 will be given of such resumption in these coht mns. 



(3w iHatt)cmatiral Column. 



w 



EASY LESSONS IN CO-OEDINATE GEOMETKY. 



By Bicharo A. Proctob. 



{Canlinued from p. 19.) 



E shall now present two forms of the equation to a Btraigiit 

 line which are often found useful. 

 52. Prop. — To find the equation to the >:traight line in terms of the 

 perpendicular let fall from the pole upon the straight line, and the 

 a iigle u-hich thin perpendicular makes with the axis of x. 



We have already seen (45) that the polar equation to the straight 

 line in terms of p the perpendicular from the pole on the line and 

 a the angle which this perpendicular makes with the initial line, is 



r cos {6 — a) =p (i) 

 that is 



t* cos Q cos a + r sin Q sin a=p 

 but transforming to rectilinear co-ordinates we have 



r cos = £ and r sin i3 = y 

 thus (i) becomes 



X cos a -Hy sin a=p 

 the re()uired equation. 



.")3. On account of the importance of this form of the equation to 

 a straight line we add an independent investigation. 



Let C A B be a straight line on which the perjiendiciJar O A is 

 let fall from 0. Suppose OA=p, and ZAOB = a. From P any 

 point in C B draw P M perpendicular to X, and draw M K parallel 

 to B C, and PN parallel to -\ K — that is, perpendicular to KM. 

 Then 



OA = OK-^KA = OK-^PN = OMcosa■^PMsino; 

 that is 



X cos a+ y sin a =p, 

 as before. 



51. As an illustration of the mode of obtaining any one form of 

 the equation to a straight line from another form, we shall show 

 how the equation just obtained may be deduced from the equation. 

 X y 



a ^ ' 



in tcnns of the intercepts a and b. 

 In 53 (using the same figure) and 



o=OB=j) sec CI ; and 6 = C=p cosec a ; 

 substituting these values in (i) we get 



y 



= 1 



p sec a p cosec a 

 that is 



iT cos a + y sin a= p, 

 as before. 



55. Note. It is necessary that the student should carefnltv 

 observe in what manner equations of the form just obtained are 

 to be interpreted. And first of the perpendicular p and the angle 

 a, it is to be noted that a positive distance p is to be measured along 

 a line inclined to OX at an angle a (measured as described in 

 former articles. The required line is then to be drawn at right 

 angles to the line thus obtained, through its extremity. Thus in 

 the equation 



a: cos a + II sin a — p = 

 p is invariably a positive quantity. 

 5t>. To put the general equation 



A.T-I- B(/-hC = (i) 

 into the form just obtained, proceed as follows : — Divide each 

 term by v^A- + B" Ki'''''B that sign to the radical which w^ill makp 

 the constant term C negative. Thus (i) becomes 



