• KNOWLEDGE • 



[Aug. 4, 1882. 



tvn> changes tsildig pUce in ii,y, :, when » is cliini,'>'d into 

 »+A--. Then, 



u+iiu=fy+ Ay) (: + &-) 



— y2 + «Ay + !/At+ A:A? 

 au'I snbi-racting former from latter 



Au = iAy + yAt+ AyA» 



and 



A: AyA< 



AS 



Now when Jx is mmie imlcfinitely email, this becomes 



— • : — ^y — + — :-. (an indelinitelv small quantity) 

 dx dc dc dr 

 dn dij dt 

 ""' Ti^'H^^dJi 



AnJ in like manner it may be shown that 

 if u-.-iryj .... 

 du dv . die du dz . ,,, 



dx dx d r dx ' dx 



III. Let u = ^, 1/ and : bcin«- functions of x. Then — 



= -^1 +±!l\ A-J^ + terms involving (A;)' io- ) 

 ^y Ay y.A; 



+ iterms involving Ay.Az, A;', ic.) 

 .".. •subtracting, A u = — : — 'Ll — f + (terms involving As', Ac.) 



involving As', &c.) 



and £^=.i.^_-l'.£i + (te 

 Ai ; Ai 2' Ax 

 so that proceeding to the limit, 



dx" z dx :' dx (C.) 



Take next the function of a function. Suppose, for instance, 

 thatu-^(y), where y is a function of ». Lot y change to y + i y 

 when Ay is some small but finite increment, and with this change 

 let u become u + Au, and x become i r Ar. Then wo have 

 u + Ju = ^(y + Ay) 



Au = 9(y + Ay)-#(y) 

 and Au ^(w + A«)-«(i/) 



Ai~ '' J^ 



^^(y + A y)-»(y) Ay 

 Ay 'i* 



Now let Ay become inde6nitely small, Au and Aa becoming also 



. , _ . a(m + Aw) — 0(i/\ du 



lu'lefinitely small. Then ^ =-•' becomes t", in accord- 

 Ay dy' 



ance with our definition of a differential cocflicient, and tho above 

 equation becomes 



du du. dy 



dt~ dy ' dx (T)j 



If we note that relation (C) may bo written thus, 

 d« !_ dy dn 



di" z ' dx'^ y dx 

 fa relation not apparent until D had been established) we see that 

 relations A, 15, and C may bo combined into the first of the following 

 rules, while D gives tho second :— 



Kulo I. — To deOrmiw the differcrilial coefficient nf a enmponitc 

 /unctinn of a vnria},U, with rftper.t to Ihin variahle, differentiate each 

 component fiinrii'.n with renpect to the variable, aa if the rest were 

 eo^$tant, and add the rebuilt. 



Rule II.— To determine the differential coefficient of a function of 

 a function of a variable, v.iih ntpeet to ihif variable, differentiate 

 with reipcct to the lael-menlioned function, and muitiply the 

 rerult by th« differential coefficient of thia function with reepect to 

 the variable. 



It is hardly neoeosary to note that the differential coefficient of a con- 

 rtani it zer.\ for this is only another way of saying that a constant 

 does not vary. It is also clear that if the differential coeffu-iml of 

 a quantilu ui zero, Ve quantity muit be cnnntan', for this is on!y 

 saying that a quantity which does not vary is constant. It is, more- 

 over, independently obvions, but comes out directly from Rule I., 



that if the differential coefficient of a quantity I's known, then the 

 differtntial coefficient of the quattity multiplied or divided by a 

 constant »s the former differential coifficient mu((«^Jied or divided 

 by the same constant. 



Wo had occasion some time buck to note that (no q<iantilies 

 which have the same diff'rmUial orfficient can only differ by a 

 constani quantity. Wo can now prove this. For lot there bo two 

 quanlitie.-i y and t, both functions of a variable a', which have the 

 Eamo coefficient with respect to j ; so that 

 <ly^^- 



jx " tic 



Then if v-: = t', we have, by Kulo 1, 



dx dv dx 

 wherefore u is a constant,- i.e., sinco u=y — 

 from 5 by a constant quantity. 



In our ne.\t wo shall give some examples of diiTerentiation by 

 these rules. 



only differ 



I'lioBLEM 30. — A solution which has been snggcstcd to mo of 

 Query No. 30, page 323, vol. i., I give below i— 

 »• -h y = 11 

 y' -f X = 7 

 ','. x' + y' + X + y ■= IS to this add and eubtracl 2i'y 

 i' -^ !/' -I- 2iy + J -(• y = 18 + 2xy 



2xy 



o' must bo a square or ( 



30 being a square number i 

 must equal 1 ; wliereforo s + 1 = 6.— P. 



[ Why does it follow tliat bocanse «' + 2.s- -I- a' is a square. 

 a' = I? •!» -h 2 X 4 + 5« = 49 = 7^ andj those arc all whole 

 numbers : x and y might be fractional. — Ed.] 



A mathomaticol correspondout (Sf r. V. Cowloy) asks mo to ex- 

 ])lain why, after saying, at p. 101, col. 2, that at tho beginning of 

 the ith interval the velocity is (r— 1) gr, and at the end of that 

 interval the velocity is njr, 1 go on to sny that tho space described 

 in the interv.il lies between (r — 1) gr' and rgr-. He says this seems 

 to bo assuming what I liavo to prove, " though in reality the space 

 so traversed is in reality J rgr'." There is no assumption in tho matter, 

 however; the spaco described in a given time, t, with uniform 

 velocity, v, is vt ; so that the space described in time r, with velocity 

 (r— 1) yr is (r— 1) jr', while space described in time r, with velocity 

 lyr, ia rj7r»; wo know that the space described in tho rth interval 

 (rin length of time) lies between theso values, because tho velocity, 

 is never less than (r — 1) yr during the interval, and never greater 

 than rgr. Our correspondent scorns to confound in some way the 

 spaco described at the end of tho rth interval with tho space de- 

 scribed diiriTiy the rth interval. The former is of conrse - (rr)" 



(by tho usual formula «=!./"('. 

 interval is in realitv 



The space described during tht 



![(')•- 



that it 



^(2r-l)r 



Writing this g {r—\)r^ wo see that it lies, as a mutter of fact, just 

 half way between the two limiting viilues mentioned above and in 

 tho text. In most cases where limits are mentioned in this Way, 

 all that wo can assert at first is that tin; quantity dealt with lies 

 somewhere between the limits mentioned, not that it lies, as it 

 happens to do in this case, exactly midway between thom. 



Tlio same correspondent asks how, after showing that the spneo 

 traversed lies between two qnan(ities, I can rensonably say that in 

 the limit these two quantities are eipial. "How can a quantity 

 lie between two equal quantities.'" I fear if Mr. Cowloy finds 

 difficulty here, the whole subject will appear very perplexing to 

 him. Of what use would it bo to show that a quantity which we 

 want to determine exactly lies between two quantities which 

 ultimately differ by a finite quantity ? If wo can show tliiit a 

 quantity always lies between two others, H and C, which may be 

 written A + a and A —a', while both tho quantities a and a' may bo 

 made, under certain conditions, less than any assignable quantity, 

 then, and then only, we rian sny that under those conditions X - A ? 

 Why should we find any difficnlty in the circiimstanco that under 

 these conditions B — C — A ? 



Piv>iir,KM.— To integrate ' 



\^y 



