154 



• KNOWLEDGE 



[July 28, 1882. 



- Ac, can only bo regarded as correct wlien the series on the right is 

 convergent. It is not so when * is greater than unity. The real 

 eqnation is — 



the last term of which can be neglected when x ia less than 1 nnii ii 

 is taken indefinitely large, but can br no means be neglected when 

 j: is greater than 1. For instance, when J = 2, this equation 

 becomes — 



jJ^=_l = l + 2 + 4+ +2--2— 



when? the last term cjccceds in absolate valno by unity all the other 

 terms taken together. — jKs.>iiE BoycE. Not the ellipticity nlonr, 

 bat that and inclination of sun's path to ecliptic explain the 

 obfen-ed |KCuliarity the so-called equation of time. It is not 

 easy to explain tho peculiarity without diagrams. May ono of 

 these days lind space for an illustrated account iu these columns. 

 — F. C. Geee.v. The explanation of the tides to which you refer, 

 rejieated so often— given not only by Mr. Chambers and others 

 ii)i:-mathcmaiical, bnt by Mr. Christie, tho Astronomer Koyal, — is 

 entirely incorrect. It may bo compared with this, as an explana- 

 tion of the reeling of an inclined top, — " Gravity acting on 

 an incline<l top, causes it to topple over," the thing to 

 be explained being why the to[) doe.'f not topple over. 

 Ocean Tides will form a good subject for a future paper 

 in KxowLEiiCF. but though Time and Tide wait for no 

 man, those tides must wait for Time. Just now we are tied 

 to time and space. — W. Cave Thomas. Thanks ; will try to find 

 room.— A .Sceptic. Scepticism of that sort has done more to 

 foster Buperstition than tho stupidest credulity. Take mesmerism 

 as an example. There is something in tho phenomena observed, 

 though not what charlatans claimed ; the credulous accept pretty 

 nearly all ; bnt the foolish sort of sceptic runs about proclaiming 

 there is nothing in it : ho is shown that unmistakably there U 

 something in it ; forthwith he runs about proclaiming, just as 

 foolishly, full faith in all the absurdities claimed by professional 

 mesmerists. If yon want not to bo led astray by the untruths 

 told about Tlionght Rc.iding, do not mislead yourself by 

 tho supposition that there is nothing to bo explained. Meii 

 like Carpenter and others, who have done more than any 

 to take the ground from under tricksters, in psychological and so- 

 called spiritual matters, may be much more safely followed iu what 

 they admit, than tho vulgarist you quote in what (knowing notliing) 

 he rejects. Heal science has never encouraged a spirit of Incredulity, 

 though placing Douht in the chief place among the virtues. Show 

 me a man thoroughly incredulous, and I will show yon (ivithout 

 going very far) one who is on the way to l>o most thoroughly de- 

 ceived. 



(Jhir i¥latt)fmatical Column. 



EA.SY LES.SOX.S IX THE DIFFI- RESTIAt, CALCULUS. 



No. V. 



By Rkiiabd K. Proctoe. 



IT may be well to apply the method already need to determine tho 

 differential coefficient of sin x, to tho other simple trigono- 

 metrical functions ; for rot only does the method indicate well what 

 a differential coefficient is, but as all these functions are dealt with 

 in the same way, the memory is aided to retain all their diifcrentinl 

 coclTicients. 



As l)cfore, wo nso the circnhir measure for angles, and regard 

 ladiiu as unity. The construction of the Hgnrc is obvious. 

 Pat arc A15-r, B6-.a,r. 

 Then wc have seen that if 



,h, 

 y~anx; j^-coga, 



I'ut now y — cos »=0m, — radius Ijoing unity. 

 Ay-cos (x + A*)-cosz-Oik-Om=-;./rt 



Ai" ¥!>" bI' 



n t, 'B B; 



■■.nil, we have — — _ 



and when j 



ind 



Ixilli indefinitely 



•'. when y — cos » ; _i — — gin r. 

 dx 

 Tut next y ■> tan * — AT. Doscrilx- arc Ti about •") 

 Then Ay - tan (z + Ai-)-tan j-rA-TA-T(. 



Ax~B6''Ta' B6"t»'Bo' 



en A I and A y tiro 



bot li indefinitely small, we have — = — = 

 Tn Out 



, T(i to o .1 . , 



and — = — =scci'. So that when 

 BO t)A 



y=tnu 



When y=cot x, ivc have a precisely similar construction, but \\v 

 may here conveniently use the samo construction which has given 

 us tan X to give us cot ,r. For this purpose we set OC at right 

 angles to OA and call tho arc C!), x ; and Mi, A.r. Then we get A( 

 for tan x and AT for tan (ai + Ax). So that we havo tho sttnio 

 difference Tf, only in this case it is a diminution instead of an 



TO TO 

 increase. Wo havo also, ,j7;=7tT = sec BA = cosecCB='ioseOK 



BO~OA" 



(when Bl) vanishes). Her.co 

 dy 



y — cot c 



dx 



= — cosoe'J) = — 



Next take y = see «! = 0T. 



Ay = sec (a; •»■ Ax) — sec x 



Ay_ nt n 

 ' Ax~B6"f 



nT AT 



and when Aa; and Ay arc both indefinitely 



sec IS J — =tan ic . BOO a' = - 



cos'aj 

 as in dealing with cot., 

 C0800 X «= —-:— 



And similarly, cilling CO!), 



y-coseo x; ^= — i 



dx sin-x 



Tlio student should havo tho differential co-eflScients of tho six 

 simple trigonometrical fimctions at his fingers' ends, as they are 

 constantly wanted. Tho six functions are here collected into tho 

 three which havo positive differential coefficients (for arcs less 

 than a quadrant), tho sine, tangent, and secant j and thoso which 

 havo negative differential coefficients, the cosine, tho cotangent, 

 and cosecant. 



dy 



If y - I 



y - tana:; ^ = , 



y = sec a: ; 



cos's 



If M = COB ; 



= — 81 



Lot us next try tho invorso trigonomotrical functions, and see 

 whether they admit of similar treatment. 



Take y — 8in~'x. Then in our figure wo may reprceent y by arc 

 BA, whose sin is Tim, so that Mm roprescnta x 

 Thuay-BA x=Bm 



Put U= i^xBO that (»■(• Ax) -oA 

 Then Ay = sin"' (it -f Ax) -sin"' Ax 

 = arc Ai-arc AB 

 - Ub 



