102 



KNOWLEDGE 



[Jdly 7, 1882. 



Tlje arpa, then, representing the spncc actunlly traverscil lies 

 somoirberc between the two rectangles P N and iii K. Thus the 

 total <;.;uf .liMrilcil in time ( is reprcseiitcil by nn arcn loss than 

 t!.i \ It, B C ami theouttTitigiaj; Adildec/, Ac., and 



pn . Utwocn A H, 15 Cand tlie inniT zig/afTddt't'c'/, 



ii -^ r friiiM each other by the sum of the small 



I'.i: , c-f', Ac. Now, altiiough when ii ia made 



vir fore each rectangle like i.ni very small, the 



iiu^ wl rectangles is very great, this by no means 



Jir> 1 .lining very small. For sliding the rectangle 



•».. , AB to the position pq, and doing the like for 



all iL.- suiiiU i-;irullelo(jranis, wo have them finally covering the 

 rectangle Ct ; and the area of this rectangle becomes as small us we 

 please, when, n being luken as liu-gc as we please, kH becomes as 

 small as we please. Therefore the area of the triangle ABC, which 

 lic« between the two just named, may bo mado to differ from cither 

 by an area less than uny that can bo assigned ; and this is tho samo 

 as saying that this area correctly represents tho space fallen though 

 from reit under the action of grnrity. Now AB = f, and 110" = ;)'. 



Uence area ABC-?i'- And s=^" 

 2 2 



[Tlic reader must not fall into the mistake of supposing that wo 

 are here equating a distance to a triangle. AB n'prescnts a period 

 of time, and BC n velocity, — in this way only, th;it the number of 

 nnita of length in AB are supposed to correspond to tho number of 

 unit* of time in t, Ac. Just as we get tho right number for the 

 ■qaare feet in a rectangle by multiplying the numfccrs representing 

 tho lengths of the sides, thouirh feet cannot bo multiplied by feet to 

 give square feet, so here we get a correct numerical result, by using 

 the lengths of lines to represent numbers.] 



Now, in each of the above processes, we have employed an 

 artifice, algebraical in one case, geometrical in another, to obtain 

 a result such as the integral calculus obtains in a systematic 



We know in this case that the velocity at tho end of any time / 

 is g t, or, in other words, that tho rate at whioh tho space traversed 

 ia increasing, after time / has elapsed from rest, is j t. Now, wo 



showed in Paper I. tiat tho differential coefficient of ^ wiili 

 respect to f is j t, explaining this to mean that the rate of increase 

 of the expression 2— with the variation of t is represented by g t. 



Supposing, then, thot, in dealing with tho fall of a body under 

 gravity, we know velocity at time t from rest, — or the rate of in- 

 crease of t, the space traversed, at this time, — to bo gt, and also 

 know that gl is the differential eoefficicnt of — — with respect to 

 'i — we can at once write 



« = -^ ± some constant. 



(It has not, indeed, been yet shown that two quantities can only 

 hare the same differential coefficient when they are either equal or 

 differ by a constant i|uantity j but for the present this is assumed, 

 aa wo are now only illustrating tho nature and nso of differential 

 coefficients, Ac.) And as when 1 = 0, » = 0, the constant must bo 



zero. So that we have, simply, «-.£- . 



In other words, from our knowledge that — is the quantity having 



nl for its differential co-efficient with respect to /, wo are at once 

 able to say that a falling body whoso velocity at time t is gt, will in 



time t from rest traverse tho space £-. Tho notation employed for 



■'■'■ ■■■•.». Note tliat what wo have to determine is 



"•' ' - described in the infinitely small intenals of 



*'" ■■ liolc time ( is supposed divided ; in other words 



»'■ ■' "f all small spaces, each represented by jt.At 



when Ai i« ti,, ,i,,remtnt of the time. This sum, which we should 

 represent by igl.M, were the number of spaces not infinite, is in the 



integral calculua represented by pjl.dt, when the nnmbcr is infi. 



nitely great and M infinitely small, and wo write 



gt.dl ■= : — + a constant. 



/" 



How wc find tho sum corresponding to particular limiting values 

 of the variable will be shown farther on. 



Of coarse, wc have here selected a case where wo know 1>ofore- 

 hand that quantity of which the riropic expression we were dealing 



with is tho differential coefticiciit. In other cases, wo might hnvo 

 more or loss trouble to determine this ; but a great number of 

 diffeivntini coefficients are known, and in every such cnso wo can 

 at oneo write down the cxpi-e.esionsof whieh they uro tho differential 

 coefficients — or, in other wools, wc can at onco solvo our jirobleni. 

 For other cases there are methods by which cither an exact or 

 aj>pro.\inuito solution can be readily worked out. 



'I'o sum up the elementary points thus far illustrated : — When 

 there is ii iiiiinitilij lehose value dependu on some variable, the 

 diferenlial cocili ii'iit of the quantilij with respect to that variable 

 represents the rate at u-hich the quantity varies at the variable 

 changes in value. 



When ifi) know the rate at which a quantity depending on some 

 variable chanjes with change of the variable — in other u-ords, when 

 we know the differential eocfficietit of tho quantity vith respect to 

 that variable — we can determine the quantity itself, if only we Itnon- 

 what quantity it is which has that differential coefficient. 



When wo have indicated how tho differential coefficients of n 

 number of expressions can bo dotcniiincd, tlio impcirtniico of thesn 

 points will be recognised. 



(ro be coutinuul.) 



Pkoblkm 11. — AVo prefer to solve this problem in the followi 

 general form, us more instructive than a special casi : — 



Let AC, UD, the radii of two drums, he respcdirelj a and_ h 

 (b > a) ; and let AB = d, required Ihv Icnjth if belling necessary for 

 thern. 



Let OCD be a coinmun tangent luretiiig BA piodueod in d. 

 Then, 



()A 



: p tay 



i:c = a si.-' (-) 



He-jco lenglli of biltin,' rcqui ed = :J (aic EC + cD + are DF) 

 is kncw.i. 



TiiK Poisonous Constituents op Tobacco-smoke.— A scries of 

 experiments has been recently conducted by Herr Ki.ssling, of 

 Bremen, with tho view of ascertaining tho proportions of nicotine 

 and other poisonous substances in tlio smoke of cigars. His paper, 

 in Dingler's Polytcchnisches Journal, gives a useful r(?sum(! of tho work 

 of previous observers. Ho specifies, as strongly poisonous consti- 

 tuents, carbonic oxido, sulphuretted hydrogen, prussic acid, picolino 

 bases, and nicotine. The first three occur, however, in such small 

 proijortion, and their volatility is so great, that their share in 

 tho action of tobacco-smoke on tho system may bo neglected. 

 Tho picolino bases, too, arc present in comparatively Rniall 

 quantity; so that tlio poiscmons clinractcr of tho smoke 

 may bo almost exclusively attributed to tho largo proportion 

 of nicotine present. Only a small part of tho nicotine in a 

 cigar is destroyed by the process of smoking, and a relatively 

 largo portion passes off with tho smoke. The proportion of nico- 

 tine in tho smoke depends, of course, essentially on tho kind of 

 tobacco; but the relative amount of nicotine which passes from a 

 cigar into smoke depends chiefly on how far tho cigar has been 

 smoked, as tho nicotino-contont of tho unsmokcd part of a cigar is 

 in inverse ratio to tho size of this part, i.e., more nicotine tho 

 shorter tho part. Kvidcntly, in a burning cigar, tho slowly- 

 advancing zone of glow drives before it tho distillable matters, so 

 that in tho yet unburnt portion a constant accumulation of tlioso 

 takes place. It would appear that in tho case of cigars that are 

 poor in nicotine, more of this substance relatively passes into 

 smoke than in tho case of cigars with much nicotine ; also that 

 nicotine, notwithstanding its high boiling point, has remarkable 

 volatility. — Times. 



