366 



KNOWLEDGE 



[Oct. -11, 1882. 



liUd TOO found an explanation ; it will interest many who know the 

 canine dental formula, but not the actual sha{<o of cuuine jaw 

 bonfs. to learn that the second and thinl premolars of the npper 

 jaw have each a double socket, to each prong a socket, llnvo sent 

 the hieroglyphs to Miss EdwonU, but cannot, as a rule, expect 

 ].rvi;. i:is h. !v.il (!. C. Ckofpir. Kuther a widoqnestion. There 

 .Titary books on mothemntics written to meet so 



■.ts. -Ckai'ATIM. You will tind that Airy fully 

 further on in his Tracts. The inclination of the 



. increases, owinj; to the solar attraction, as tho 

 1 an equinox to a solstice, and continually 



pusses from a solstice to an equinox. This is, in 

 : -"hir nutation. — S. C Tho momentum of a body 

 riintion in it, and is representtxi by the product of 

 :''..<• 11 ..-< II.-... lie velocity. The vis viva of a body is represented 

 l.v the product of its moss into the square of its velocity. — JiLES 

 MniNV. Thanks for your pleasant comments on my lecture at 

 \Vijtl>oame-park Institute. As to your queries (1) 1 meant only 

 :liiit 1 he centre of Jupiter is denser than those outer parts which 

 .ii-i' formed of deep atmospheric layers laden with clouds. (J) Tho 

 i.'ise and uproar going on in the solar atmosphere would not bo 

 iieani beyond its limits. I said " if we could visit the sun and 

 livi-" we should hear such uproar. (3) Tho water on tho 

 moon would gradually be withdrawn into the moon's substance as 

 the contraction duo to cooling left capillary cavities for it. The 

 atmosphere would partly follow tho retreoting water, partly be 

 gradually nxiuced by entering into chemical combination with tho 

 substances forming the rock surface of tho moon. — X. Z. Have 

 received no answer to that query ; will certainly publish when 1 do. 



KLECTKICAl,. 

 Pimxix. I regret to say your question is so mixed that I cannot 

 ailequately answer it hero." Brielly, to work two electric lamps 

 (presumably incandescent) yon would require twenty-five to thirty 

 Uunwn cells, or a dynamo machine, which would require more 

 power than o man could exert, tn drive it. Large steam-engines 

 fur driving dynamo machines light nlx>ut eight lamps per liorse 

 I>ower. Sec previous articles in K.nowleuge on the subject. 



evix iHatljnnatiral Column. 



EASY LESSONS IX TUE DIFFERENTIAL CALCULUS. 



By RicnABD A. Proctob. 



No. XI. 



THE reader has seen enough of tho applii 

 calculus to problems of maxima and n 



tion of tho diffpreiitinl 

 nima to feel satislied 

 of the value of tho method. I may now briefly consider another 

 class of problems to which tho calculus may be conveniently applied. 



A differential cocfBcicnt is in reality a fraction of the form -, or 



O' 

 what is termed a vanishing fraction, and like many other vnniBhing 

 fractions it has a real value. Now it is often necessary to find the 

 value of vanishing fractions, and though orrlinnry algebra miiy often 

 be successfully applied for this purpose, it is not always possible, 

 and often, though possible, it is exceedingly difficult, to evaluate a 

 vanishing fraction in this way. 'llic differential calculus cnublcRUH 

 to treat sufh fractions very simply. 



Let US take such a vanishing fraction, and consider what is rctdly 

 required for its evaluation. 



Take the fraction . 



the 1 



crator and denominator of whicl 



Ujth vanish when x is equal to a. Now we can at once find the 

 value of this fraction by striking out the common factor x — a, an<l 

 so changing it into the form 



the value of which is 



rhcD 



Even to this simple 



'IT'I 



cation of algifbra there is an <>bje<-tion ; since striking out a factor 

 equal to i« a questionable proccsM. The result, however, is correct 

 enough. 



But the only Ictplimatc way of treating such a fraction would be 

 to inquire what its value is when x is taken very nearly equal to u, 

 as a -^ /i, and so trying to find out what value thcfraction approaches 

 to when « is exactly opial to a. Ixjl us do this. Our fraction 

 becomes i«'io.)"i,ooiK«^ ~ 



(o-t-/.;' 



3a' h»3a A'-t A' 3a' -f! 



,h-,W 



Here wo can seo nt ouco that by making h small enough wo can 

 pet our fraction os near as wo please to 



au' (' 



anil we therefore conoludo that 



when h i;) 



0, or X equal to «. Hut a little consideration will show tho reader 

 that tho process corresponds exactly to that for obtaining tho 

 differential coefficient both of tho numerator and denominator. 

 Hence he will be prepared to find that when we have a fraction of 

 the form -, where both the expressions u and v involve .t, and both 



y 



vanish for a certain valuu of x, the fraction may be evaluated by 



siniplv writing for u its diCfereiiliiil coeflUient ' ", and for y its 

 dx 



difTereiUial eoellicient -", 



Tuke. for instance, the expression 

 x-\ 



which assumes the form - when m- 1. Following tlie rule, we write 



for a new numerator tho differential coefficient of a-— I, i.e. 1, ami 

 for a new denominator tho differential coetlieieiit of .c' — 1., i.e., Gx". 

 Our fraction thus becomes 



But it may happen that the new fniclion tluis formed is itself a 

 vanishing fraction. In this case we must repent tho process until 

 wo obtain a fraction which is not indetcrniinute in form. Tliu.'* 

 suppose wo have tho fraction 



{x-\y _ 



a-'-ax'-l-aai'-yjj'-f 3j-1 

 which is of the form - when o; = l. Wo apply tho rule, getting 



■ 3(x-l)» 

 0.(^-15x«-i-12*'-Gr+:t 



which ia still of tho form -. Again, applying llio rule, wc get 

 _3.2.(;r-l) 

 30.c*-U0»'i-3(ij:»-U 

 which is still of tlie form -. Lastly, applying the rule yet ouco 



more, we get 



.1.2.1 

 120j:'-180x'-(-72x 

 And when .r — 1, this fraction has tho valno 



C_ j^l 



]20-lH0-l-72"l2~2 



Other vanishing fractions may bo similarly treated j and this 



application of tho differential calculus thus becomes of great utility. 



©ur fflaaijigt Column. 



By " FivK Of Clubs." 



IE Editor sends me the following game, played on tho 

 opening night of tho Kew Whist Club. It is a very imple 

 of bringing in a long kuit : — 



A. Tnn Hands. 



C'/u6»— Q, 7, 6, 4, 3. 

 Diamonds — A, Q, 7, '• 



4,2. 

 Spades— t). 

 Uearts— .V. 



Diamonds — Kn, H, 3. 

 Spades— H, 7, 2. 

 Heorts— 10, 8, 7, 4, 3 



r. 



I 7,<l>,— 10, 6, 2. 

 Diamonds- K, 6. 

 S|mdeB-Kn, 10, 4,3. 

 Il.iirlK-Kn, 9, G, 5. 



'•/„/.«- A, K, Kn. 

 Diamonds- 10,9. 

 S|.a<l.-8~A, K,Q,C,6. 

 Diarls-K, Q, 2. 



