whence by addition, and by reason of equation (75), if x'=x, y = y, z1 = z, we have > (£) (|) + «■-«> (£) = °= TM a result the same as (76). In the preceding process the relation between the partial differentials of the functions in (77) is found, when the problem is resolved into its most simple elements; an abridged form of it is more useful in application. And as the result thus obtained is applicable to other cases, it is desirable to insert it. Let the functional form which occurs in the y- and the z-partial differentials of (77), as they arise from the variation of the first subject of the function, viz. ^—-, be represented by ?i; and similarly let r2 and F3 represent those which arise from the second and the third subjects: then id¥\ z—c 1 idr\ 1 x — a \(h/> ~ J^c Fl ~ {y-bf Fs' ••• <*->(£)+<*-»> {£)-* Ex. 3. As another example of a similar process let us take the function ?_£=/!_?); (80) a c Ko c' c \dyl ~ \b c \dyiy ^b c>' •;■ «OW£) = And if (80) is put into the form ix z y z\ u = F( , f--), by a process similar to that of the last example we shall find •£)+ »($ + •©-■>' m and further, if (80) is put into the symmetrical implicit form, ly z z xx y\ « = r I •{ , , '); (83) sb c c a a b' and if, as in the preceding example, Fi, F2, F3 represent the functions which enter into the partial differentials, according as they arise from the differentiation of the first, second, and third of the subjects, we have CHAPTER III. SUCCESSIVE DIFFERENTIATION, AND THEOREMS Section 1.—On successive differentiation of an explicit Junction of one variable. 54.] In the former part of the last Chapter rules have been constructed for differentiating explicit functions of one variable, and thus of deducing from f(x), which was assumed to be the typical form, d.f(x) or f'(x) dx; in this case x was made to increase or grow by the infinitesimal dx. Suppose now that x is increased again by an infinitesimal variation, and that it is our object to inquire what effect such a second increase will have onf(x) and on f'(x)dx; this second increment may or may not be equal to the former one: doubtless the simpler case will be when it is equal, and therefore we will consider it first with this limitation, and subsequently discuss the general case when the successive increments of x are not equal. When x increases by eqnal increments, or grows, as we may say, at an uniform rate, I shall call it an eguicrescent variable *. Or we may consider the subject from another point of view; f (x) is in general a new function of x; and therefore as it was derived from f{x), so by a similar process may another function be derived from it. This new function by an analogous notation is symbolized by f"{x), and will in general be another * The variable, which I have ventured to call Equicrescent, and thus to coin a new word for, is by most writers called " Independent," and by some old ones " Principal Variable j" to the latter term the objection is, that it does not express the characteristic property of the thing to which it is applied, and has in fact no pretension to appropriateness of nomenclature; the former term is by all writers, and in the present treatise, used in a different signification, viz. to express that variable which first changes value, and due to the change of which the other variables change, and are therefore called dependent: see Art. 12; and as such an independent variable may or may not be equicrescent, it is inconvenient to use the same term in two different senses: and especially as the term does not express that character of the variable which renders a distinctive appellation desirable. PBICE, VOL. I. N function of x, and thus will admit of having another function derived from it by a similar process, which will be symbolized by f"'(x), and so on. Thus may derivation be considered an algebraical artifice by which successive functions are formed, each from the preceding one; and these are called the derivedfunctions or derivatives of different orders of f(x), viz. /(*), /'(*), /"(*), /"'(*), /"<*), fn+\*) ;(i) f(x) is called the primitive function; f\x) the first-derived; f"(x) the second derived; f"(x) the nth derived function of f(x). It is also to be observed, that each function in the above series is the first-derived of the immediately preceding function; and that f"{x) = f{x), when n = 0. In the same way therefore as d.f(x) = f'(x) dx, so does d.f\x) — f"(x) dx, and so on; whence we have the following series of equations: d.f(x) = f\x) dx, d.f'(x) = f"(x) dx, d.f'\x) = f"'(x)dx, y (2) d.f»-l{x) = fn{x)dx. J The (he's, which are factors on the right-hand side of the above equations, are the several increments of x which give rise to the differentials of f(x), f'{x), and therefore are not necessarily all equal; but, as above, to consider the simpler case, let us assume them to be equal, so that the several functions vary by reason of equal variations of their variables: that is, let x be equicrescent; then, since y = /(*), <fy = d.f(x) = f\x) dx. And since f'(x) is a function of x, we may differentiate again; whence, as dx is constant, we have d.dy = d.f'(x) dx; and since d.dy signifies, that the operation symbolized by the character d is to be performed twice on y, and one operation to be on the back of the other, we may, in accordance with the notation of the index law, abbreviate d.dy into d*y; and replacing d.f'(x) by f"(x) dx, and writing dx2 for the square of dx, since the rfa^s are equal, we have d*y = f"(x) dx\ |