And again, let z receive another increment not equal to the former ones; and so on, whereby the following series of equations will be formed; y+dy = f(x+dx), y+2dy+d2y = f (x + 2 dx+d2x), y+3dy +3 d2y+d3y = f(x+3 dx +3 d2x+d3x), the last general expression in which group may be shewn to be true for all positive and integral values of n, by a method similar to that employed in Art. 55. Hereby are determined the several and successive complete variations of an explicit function of x, corresponding to the successive values of the variable; but the general result is much shortened by making a equicrescent. 69.] Let all the da's be equal, so that d2x, which is the increment of one da over the preceding da, is equal to zero; and similarly d3x dx = = d^x = = ... whereby the last equation of (71) becomes ... 0; Thus the distinguishing character of an equicrescent variable is that all its differentials after the first vanish. And as this condition is of the greatest importance in the application of Infinitesimal Calculus to questions of Geometry and Physics, it is good to illustrate it before we proceed to discuss its other properties. Suppose that we are considering any function of x between the values and xo, xn being the greater of the two, and the function remaining finite and continuous for all values of x between these limits; let us resolve the difference x-xo into small elements, the number of them being of course infinite, when each element is infinitesimal; let da be the type of each element. It is at once manifest that all the elements need not be equal; that is, all the da's are not necessarily equal. And if they are not, there will be an excess of one over another; that is, there will be a d2x. Neither again need all the d2r's be equal; that is, the dr's need not be equicrescent; but if they are not, there will be an increment of one d2x over another der; that is, there will be a d.d2x or a d3x. And similarly with regard to the other differentials. But if we once introduce the condition that a differential of any order shall be resolved into elements which are all equal to one another, then all the subsequent differentials vanish; and thus, if all the da's are equal, as above, d2x = d3x = ... = 0. Or again, conceive a small body, as a billiard-ball, to move over a finite distance in a straight line in a finite time; consider the straight line to be the axis of x, and a certain point in it to be taken for the origin; let the body at the beginning of a certain time be at a distance xo from the origin, and at the end of the time be at a distance x, and conceive the time of its passing over the distance x-xo to be t; resolve this time into equal elements dt, and the space into corresponding If the body moves elements, of each of which the type is da. o through the whole space at the same rate, viz. with the same velocity, then, during equal times dt, equal spaces de will be described; but if the velocity varies, equal spaces will not be passed over in equal times. On the first supposition all the da's will be equal, therefore dx = 0, and x is an equicrescent variable; on the second, the dx's vary, and dx, which is the increment of one dr passed over in a time dt, over another dæ passed over in the preceding or succeeding time dt, as the case may be, is the measure of the increase of the rate of motion. If then all the d2r's are equal, we say that the velocity of motion is continually increasing, and at a constant rate; but if der is not constant, then the rate of increase of the velocity of the ball is no longer constant, but varies according to some law on which the rate of increase depends. It will be observed, however, that if the whole time of motion is resolved into equal elements dt, the supposition of a being equicrescent is incompatible with a varying velocity. Hence too it is clear, that generally we are not at liberty to assume more than one of the variables to increase or decrease by equal augments; for in the case above, if we resolve the time into equal elements, then, in general, unequal spaces will be passed over in equal times, and we cannot consider all the dx's to be equal, and therefore we cannot make d2x = 0; and if we resolve the distance into equal parts, then, if the velocity varies, these equal spaces will be passed over in unequal times, and therefore all the dt's will not be equal, and we cannot put dt0. In general, however, we are at liberty to choose for an equicrescent variable whichever is most convenient. 70.] Let us now consider in what manner these assumptions modify the equations of derived-functions in Art. 54. In the series there given we have Now considering f'(x) dx to be the product of two variable quantities, and differentiating it as such; and, in accordance with the former notation, making ƒ"(x) dx to be the symbol for d.f'(x) and f""(x) dx for d.f"(x), and so on; we have d2y = f'(x) dx2 + f'(x) d2x, = · ƒ""'(x) dx3 +3ƒ" (x) dx d2x+f'(x) d3x, ƒ''(x) dx1+6ƒ""'(x) dx2 d2x+3ƒ''(x) (d2x)2 +4f" (x) dx d3x+f'(x) d1x, Let x be equicrescent; that is, let de be constant; whence d2x = 0, d3x = 0,' ...... ; d3y f""(x), or its equivalent is derived from f'(x), on the is derived from fa-1(x), on the same sup Whenever therefore we meet with these or similar symbols, it is to be borne in mind that they have been successively de rived on the supposition that the variable x, that is the variable in the denominator, is equicrescent. 71.] To return to equation (72), Art. 69. Let the whole quantity by which is increased be finite, and be h; so that ndx=h; and therefore as dr is an infinitesimal, n is an infinity of the same order, Art. 8; whence we have and as de is infinitesimal, we must omit in the several numerators the infinitesimals which are subtracted from the finite h2 h3 f(x+h) = f(x) +hf'(x) + 1.2 ƒ" (x) + 1.2.3ƒ" (x) + ... a series known by the name of Taylor's Series, having been discovered by Dr. Brook Taylor, and first given in his "Methodus Incrementorum" in the year 1715. It is however of the utmost importance that the conditions and extent of its applicability should be accurately determined, and so another and more exact proof will be given hereafter; and the above may be considered in the light of a presumption, that such a relation as equation (76) is likely to be true. 72.] In certain cases wherein a derived-function becomes a constant, and therefore the subsequent derived-functions vanish, the number of terms of the series (76) is finite; but generally the successively derived-functions are functions of x, and the series is continued to an infinite number of terms; but the sum of all the terms after the nth may be expressed as follows by an algebraical formula, and thus proved to have a value contained within certain determinable limits: since h h3 f(x+h) = f(x) + {ƒ'(x) + 12ƒ” (x) + ... + hn+1 therefore the sum of all the terms after the nth is equal to {some quantity > ƒ"(x) and < f1(x+h)} ; (79) the latter factor of (78) is I say greater than ƒ"(x), because the algebraical sum of such a series of terms is greater than its first term; and it is less than ƒ"(x+h), because if equation (77) is derivated n times, there results h h2 ƒ"(x+h) = ƒ"(x) + "="ƒ»+1(x) + 1.2 ƒn+2(x) + and with the exception of the first term of this series, every term is greater than the corresponding term of the series in the latter factor of (78), because, the numerator of the fractions being the same, the denominators are severally less. Hence representing by some positive and proper fraction, that is, some number greater than zero and less than unity, we may symbolize (79) by hn 1.2.3...n fn(+0h); (80) for (80) is too small when = 0, and is too large when = 1; hence some value of 0 between 0 and 1 will give the correct value. Thus Taylor's Series becomes As the right-hand member of the equation has a determinate |