March 16, 1883.] 



KNOWI.EDGE • 



169 



falls from the manv paddlo-boxes of the steamers plying to and fro 

 is very remarkable. I was told by a captain that it is due to the 

 late heavy rains, which have thoroughly scoured the river. Is this 

 80 ? Fbeuebick J. Stevenson. 



DEAD POINT IX TniCYCLE CRANKS. 



[760] — I am surprised that your experienced tricycling corre- 

 spondent should fall into the common confusion of engine-cranks 

 with tricycle-cranks. He says [Vol. III., No. 68.] "Each foot 

 controls a pedal over little more th.in one third of a revolution." 

 But the action of the knees and ankles and the friction of the sole 

 on the pedal all make the analogy of a piston-rod crank quite mis- 

 leading. 



Remove the chain or " intermediate gear " of a tricycle, so that 

 the crank runs freely without driving anything. Sit on the seat, 

 and you can turn the crank either way, fast or slowly, stopping or 

 starting at any point in the circle, and deriving no help from 

 " momentum." Except on an ascent, or on very soft or lumpy roads, 

 a tricycle can be started at any point in the turn with a very little 

 practice. There is still a weak point, but no dead point. 



D. M. 



ST.VYS. 



[761] — I can quite conBrm the statement of Mr. Bchnke as to 

 the influence of the stays on the breathing capacity of the chest. 

 \8 one instance of many — I this week saw a young lady who, her 

 height being under 5 ft., should have a vital capacity of about 125 

 cubic inches, who could not breathe into the spirometer more than 

 78. On removal of her stays, which were by no means tightly laced, 

 she at once, and with the greatest ease, breathed 118 cubic inches. 

 I am happy to add that she was so struck by this fact, that she 

 refused to resume the corset, but carried the article away with her, 

 wrapped up in brown paper. Lennox Bkowne, F.R.C.S. 



CHIRPING SPIDER. 



[762] — I fear my langnase was inaccurate, and misled " S. H. 

 W." [737]. I am quite familiar witli the cricket's provoking 

 "tweet," and I have heard the " Death Watch's " tap unscathed ; 

 but my spider made a note like the scratching of a little finger-nail 

 on a linen tablecloth, or the cutting of hardish paper with a paper- 

 knife, in short notes repeated at the rate of from sixty to eighty a 

 minute, pretty regularly. I am convinced he was a spider, and I 

 think my letter (Vol. III., No. 60, 7t)0) shows I took care to make 

 sore. But I asked a friend to come and help me to catch, 

 examine, and describe the creature. The chirping ceased before he 

 oonld come, and on looking I found the web empty and trans- 

 parent, without even a cast skin left behind. 



My landlady, once a housemaid, assures mo she has heard big 

 spiders emit a sort of soft hissing sound. D. SI. 



LETTERS RECEIVED. 



A letter (on an exegetical subject) has been sent to Editor to be 

 forwarded to Mr. E. L. Garbctt. We have not Mr. Garbett's 

 address, bnt will retain the letter till sent for by him.— Jas. Watson, 

 jnn.— H. .1. M. Untenable.— E. P.— .ilf. Peters.— M. R. Name 

 not Sir G. B. Airey. — R. M. Subject scarce suitable. Are not 

 (raspenders alwavs replaced bv belt for active exercise ? — A. J. 

 Dove, C. W. Bo'wnie, H. J. Bell, H. II. Higgins, G. G. Warren, 

 Scot, J. Abbot, B. A., J. J. Martin, Rev. Urban Smith, and 

 others. Only the profanity of certain prayers objocted to ; 

 not prayer in general. Some of you seem to think none 

 but Christians ever pray; but Jews, Mahomcdans, Buddhist!!, 

 all sorts and conditions of men, in fact, pray energetically. — 

 L. Gordon Rylands. — A. M. S. — J. Samnel. More rotations on 

 outer curve, except by crank-connected wheels, whose rotations 

 must be the same in amount, wherefore in their case there is slip. 

 — W. H. S. Monck. — E. A. You arc probably right ; but some 

 would object to subject. — T. F. R. Subject to be treated shortly. — 

 Mrs. JiiS. .lackson. — C. T. Grav. Yon mistook last vertebra for a 

 sting.- A. J. Mott.— T. PrestonBattersby.— M. A. R.— Bios. Theory 

 of planet formation quite misunderstood by yon ; planets probably 

 not thrown oft from sun. — W. B. — Upsal^. — A Country Clergyman. 

 I prefer a reflecting telescope, but others like refractors best ; of 

 the two you name, the larger instrument is relatively the cheaper, 

 the smaller is the better for its size, and the handier. The question 

 would not do for discussion, as it introduces trade considerations. 



(Pur iHatljcmatiral column. 



By HicnARO A. Phoctor. 



EASY LESSONS IN THE DIFFERENTIAL CALCULUS. 



(Continued from Vol. 11.) 



THAVE for some time deferred the rest of these lessons, having 

 been asked to givo some easier subject (though, in truth, the 

 Differential and Integral Calculu.s, as I am treating it here, is easily 

 within the range of anyone who has mastered the elements of 

 Algebra, Trigonometry, and Geometry). I now continue those 

 lessons, which will presently reach their conclusion. 



In the preceding lessons, those integrals which are at once de- 

 rivable from known differential coellicients were indicated and 

 collected into a table (V"'- H-. P- ''"3), and then two simple pro- 

 cesses for integration — inti'gration by substitution and integration 

 by parts — were consideioil at p. 501. There is another process, 

 called integration by reduction, Avhichmay be considered later. But 

 I proiMso now to consider certain useful integrals which may bo 

 obtained by the processes already described. I shall treat them in 

 a rather novel manner, though there will bo nothing new in the 

 matter presented. 



Abea ov a CiRciLAE Sector. 



/ii 



Fig. 1 



Let A B (Fig. 1) be a circular arc ronnd the centre 0, and let it 

 bo required to determine the area of the sector A B, where 

 AO = rand ZAOB = a. 



Let P Q be a small element of the arc A B. Let .^AOP = 0; 

 ZQ0P = c9. 



Then if Q P be joined, triangle QOP = OP.OQ sin QOP 

 = -sin le. 



2 

 And when i becomes d 0, or is mado indefinitely small, 



sin oe = ce, and triangle QOF^t'dO. 

 Hence, since area A O B is manifestly equal in the limit to the 

 sum of all such triangles as Q P in the sector, when their number 

 becomes infinite, we have 



Sector A B = ij^r' d9 = T,Jd 0. 



Now wo know that — 



fd9 = e + C, 



where C is some constant (which need not trouble us). Therefore^ 



Sector AOB = -(a + C- oTC ) ~ g " 



That is, the area of a sector is represented numerically by half the 

 product of the numbers representing r tho length of the radius, 

 and ra tho length of tho circular arc. 



Observe that wo learn nothing new from this application of the 

 integral calculus. It does not tell us what the length of tho arc is; 

 nor is anything derived from tho calculus in the recognition of the 

 relation between the area on the one hand and the arc and radius 

 on the other. When we take a = 2?r so as to get the whole circle, 

 we finri that the area of tho circle is represented by the product of 

 the railius and tho circunirerence.* But our modern method has 



• I write thus for shortness. As a matter of fact, an area can 

 no more be represented by tho product of two numbers representing 

 lengths, than a period of time can be represented by a number of 

 acres, or the content of a solid by a number of hours. Such an 

 expression as the above means only that certain units being taken, 

 tho relation indicated exists between the numbers mentioned. Thus, 

 if wo say that the area of a rectangle is represented by the product 

 of its bides, we mean really that if a square on the unit of length is 

 taken as the unit of area, then the area of a rectangle is represented 

 numerically by the product of the nombers representing the lengths 

 of two adjacent sides. 



