Jtjly 28, 1882.J 



KNOWLEDGE ♦ 



165 



ix~bl' ^ "^'^ ^""^"^ '^^ ^^^ ^^ *'"® ^°^^ indefinitely small 

 Bb BO . 



fl — o^"^'^'^ '•/■ llenco when y=sin-'a!; "" 



1 _ 1 

 ■v/1 — sin-i/ ^1 — x''. 

 Applying a similar method for the other simple 



see y = 



functions, 



these results. He will Hnd in e7;;y cTs: on mal^^T W geTm - 

 tnoal construcfon (or examining that already made, for the figure 

 Illustrates aU the cases) that the ralne of ^ in the case of an 

 inverse function is the reciprocal of the yalne of ^ in the case of 

 the corresponding direct function. For instance, in dealing with 

 y = sin -X, we found ^^ to be ^, whereas in dealing with 

 y = sin -'*, we found £-= —. This was to be expected, when we 

 consider that to say j=sin -'.r is the same as to say ^ = sin y, so 

 VZ'.-a" ^^ '■'. ""', "'"'P'y i"t"'cbanged, in passing from the 

 considerataon of a dn-ect function to the consideration of the 

 corresponding inverse function. Xevertheless, it would rot 

 be sound at this stage of our inquiry to determine 

 ^, from this consideration only. We shall hereafter show 



under what limitations it 



may 



At present we 



assumed that 

 to recard _^ n< 



dy 



the 



expression. It is derived from £ by supposing 4y and 4j- to 



diminish indefinitely. Now, when ty has any definite value, and 



therefore hx a definite value, it ia of course true that 1^ is the 



Lx 

 /4x 



Hy the 



r:c;prccal of ~ ; but we must not assume that c/ 



value to which ^ tends with the indefinite diminution of \y and 



^■v is the reciprocal of the value to which ^ tends under the 



and Vr„f ir'- ?'' T™'^ ^^ ^«°""""e that we can treat ,?.r 

 re«ift^- "^"If^fl^^'^delin.te quantities, whereas they are in 

 rcahty mdefimte. though they bear to each other a definite relation 

 It would matter very little so far as the ix and dy of the simple 

 £ are concerned, that we should treat these as separable quantities ; 



but as we advance with the calculus wo find occasion to diffcrentiato 

 d.fforentml coefficients, and wo are led to the use of such forms as 

 d'y S?y 



dj" ' I? ' ' '' ^o\\\A bo altogether inadmissible to regard 



rfo-' and dx' of these expressions as it they were the square and 



cube of the dx in the expression ^J- . 

 dx 

 Of course when, after starting from, say, the statement 



^^■o find g = ^J, it is easy to express thi. 

 for we have x = sec y. So that 



cos y = - and sin 



V'4 



T might hero give some exam|.le3 illustrating the application of 

 the differential calculus— with the coefficients already determined— 

 to various problems of interest. But it will be well first to get over 

 so much elementary ground as is involved in the determination or 

 ruies for differentiating all expressions whatever. For then we can 

 take a much more varied range of examples than wo could by 

 muting ourselves to the application of what we have already 

 learned. It is seldom in physical questions that wc are limited to 

 simple trignometrical functions, and we could scarcely advance half- 

 a-dozen steps >vithout feeling the necessity of rules for finding the 

 clifferential coefficients of complex functions and functions of 

 functions. Indeed we sl^ould have found the use of saeh rules in 

 simplifying what we have already done, only that it ecenied well to 

 have a few examples of the process of obtaining a differential 

 coefficient directly. Otherwise when we had to consider, say, 

 !/ - sec a ; we might have regarded y as a function of cos x, and 



proceeded from y = _I_ to write do^^-n the differential coefficient. 



cos X 

 Again we might have written, for 3/ = tan x 

 y = 8ui X . sec X 



and at once written doivn ^, if wo had the rules, which I shall 



dx 

 proceed to examine and establish (as far as is necessary in such 

 elementary pajiers as these) in my next. Then we can proceed to 

 discuss a few problems which wiU not only show the great value of 

 the calculus, but also iUustrate the real meaning of what we have 

 thus far done. 



I may, however, here pause to note that the reader must not 

 allow himself to be mystified by the use of such an expression as 

 differential coefficient, a term which might seem expressly devised 

 to deter the young mathematician from the study of the calculus, as 

 implying that it cannot possibly bo of any practical use, — at any 

 rate, in simple problems. For what idea of utility docs the expres- 

 sion " differential co-efficient " convey ? and why should an expres- 

 sion be employed really belonging to a matter with which elementary 

 applications of the calculus are in no way concerned ? — the expansion 

 of sundry functions in the form of series, of which what we are 

 calling the differential co-efficient is one of the co-efficients. What 

 the young student has to bear in mind is that what (to avoid the 

 invention of new terms) we call the differential co-efficient is in 

 reality a quantity indicating the rate at which whatever function — 

 simple or complex— we want to deal with, changes with the change 

 of the variable it involves. When we consider how manv problems 

 depend on such changes and cannot possibly be dealt with unless 

 we can determine their effects and limits, the importance of a 

 calculus devised for this purpose will be at once obvious. Wo shall 

 very soon bo able to show this, for wo shall very soon have com- 

 pleted our inquiry into methods by which the rate of increase of 

 any quantity whatever, as its variable increases, can be determined. 



0UV CI)fs(5 Column. 



By Mephisto. 



rEOBLEJ[.;No. 18. 

 By U. a. N. 



White to piny and mate in three moves. 

 (An ingenious conception.) 



