u = cos.xsin. x .. du =- dx cos.xæsin.x {cos.xl cos.x

Garnier's general example is

sin.2x

Cos.x

36. In differentiating a complicated analytical expression, it will be found convenient to substitute for parts of it; thus let

.. du=dp+dq—dr.

To find dp, substitute z=(a−x)", v="/b2—x2, t=xk, w

=

where drvdz + zdv

— ^/b2—x2. m(a—x)m-1dx − (a−x).2_(b2— x2)

n

cannot be found unless the form of the function is known) and the result = du.

Infinitesimals.

37. The doctrine of infinitesimals is perfectly distinct from that of limiting ratios, and it is by no means necessary to introduce it into a treatise on fluxions; yet, as it frequently enables us to find the limit, and in other respects throws great light on the subject, we shall here endeavour to establish it upon unobjectionable principles.

Mr. Locke, in his "Essay concerning Human Understanding," has shown that we can have no positive idea of an infinite magnitude. "Whether any one has or can have a positive idea of an actual infinite number, I leave him to consider, till his infinite number be so great, that he himself can add no more to it; and as long as he can increase it, I doubt he himself will think the idea he hath of it a little too scanty for positive infinity." Book 2. eh. 17. § 16.

If then we would define infinite magnitude or infinite space, we can only define it from its negative property of being incapable of increase or decrease by the addition or subtraction of any portions of matter or space of which the mind can form an idea.

38. It follows from this definition, that if represent an infinite and a a finite quantity, za = 2; for otherwise z would admit of increase or decrease by the addition or subtraction of a.

The following demonstration has also been given of the same proposition.

=M, then multiplying by az, a + z = Maz;

suppose z to be increased without limit, then

nished without limit, and when z is infinite,

39. If a represent a finite and x a variable quantity, which is indefinitely diminished, and at length vanishes, a + x = a ultimately.

This is an axiom, and cannot be demonstrated: but it may be shown to agree with the preceding article.

For take aax, then as a decreases, increases, and if x could be increased so as to become infinite, a would vanish; but comp. z + a: z = a + x : a; and z + a = z, therefore a + x = a when zoor x = = 0.

40. If y and x are two variable quantities so dependent upon each other that, being gradually diminished, they vanish together, but in a ratio which is infinite, the last ratio of yxy is a ratio of equality.

For take za :: y: x where a is a finite constant quantity, then comp. z + a : z :: y + xy; but at that point of time at which y and x vanish, y is infinite when compared with a, and, consequently, z is ultimately an infinite quantity, and therefore (38.) %+a: z, or its equal y + xy is a ratio of equality.

Ex. As the arc of a circle decreases, and at length vanishes, its chord and versed sine gradually decrease and vanish at the same point of time, and since diameter: chord :: chord : versed sine, they vanish in an infinite ratio, and, consequently, chord + versed sine : chord is ultimately a ratio of equality.

41. So long as the quantities x and y possess any magnitude, however small, y + xy is not a ratio of equality; for if a possess magnitude, it can be diminished, and therefore z can be increased, or za is not z: they are in

this ratio only when they vanish *.

* Whether the variables ever actually attain to this ratio is a question which has been much debated, arising, as it should seem, from the imperfection of language, which, as Mr. Locke observes,

42. The last example proves that indefinitely small quantities may vanish in a finite ratio.

This axiom is the foundation of the whole science, and it cannot be too frequently illustrated; take the ratio 3x + x2: 2x + x2, and substitute for r any numbers in succession, either whole or fractional, which are in a decreasing series, and it will be seen that the ratio increases, and that it is always less than 3: 2; hence its ultimate

rather 6 serves for the upholding common conversation and commerce about the ordinary affairs and conveniences of civil life, in the societies of men one amongst another,' than to express, in general propositions, certain and undoubted truths, which the mind may rest upon, and be satisfied with, in its search after true knowledge.' B. 3. c. 9. § 3.

When the limit is obtained by increasing the magnitudes indefinitely, the limiting ratio is one to which they can only approximate, and which they can never reach; and the reason is sufficiently obvious, for they can never actually become infinite; but when they are gradually diminished and at length vanish, I can see no objection to the position, that they do actually possess this ratio at the moment of vanishing.

Sir I. Newton has been thought to hold the contrary opinion from the following passage in the Scholium: Ultimæ rationes illæ quibuscum quantitates evanescunt, revera non sunt rationes quantitatum ultimarum, sed limites ad quos quantitatum sine limite decrescentium rationes semper appropinquant; et quas propius assequi possunt quàm pro datâ quâvis differentiâ, nunquam verò transgredi, neque priùs attingere quàm quantitates diminuuntur in infinitum:' but it is evident from the context, that by rationes quantitatum ultimarum,' he understands the ratios of the quantities while possessing a determinate magnitude; and, in fact, he concludes the Scholium with a similar caution, cave intelligas quantitates magnitudine determinatas, sed cogita semper diminuendas sine limite.' Nothing can be more decisive of his opinion than the definition which he gives of the ultimate ratio. • Per ultimam rationem quantitatum evanescentium, intelligendam esse rationem quantitatum, non antequam evanescunt, non postea, sed quâcum evanescunt.'

6

If, notwithstanding this high authority, the student should be still fearful of committing the solocism of asserting that magnitudes bear to each other a certain ratio when they vanish, he may avoid the use of the terms prime and ultimate, and may define the limiting ratio to be one to which the magnitudes gradually approximate, which they approach nearer than by any assignable difference, but which they never reach.

ratio, which is 3: 2, so far from vanishing, is the greatest which the magnitudes can bear to each other.

43. In order that the ultimate ratio of evanescent quantities may be a ratio of equality, their difference must not only vanish, but it must become indefinitely small when compared with either of them.

As an example take x2 a2 and x2- ax, which vanish together when x = a, also their difference ax - a2 vanishes at the same time, but since it does not vanish when compared with x2-a2 and x-ax, the ultimate ratio in this instance is not a ratio of equality: it is 2: 1.

44. Hence arises the necessity of admitting infinitesimals of different orders.

For since 1: :: x: x2 if x be indefinitely diminished, a will be contained in 1 an infinite number of times, and consequently is contained in x an infinite number of times, or a being indefinitely small compared with x is an infinitesimal of the second order.

For the same reason x3, x+...x" are infinitesimals of the 3d, 4th... nth orders.

Cor. 1. If an infinitesimal is multiplied by a finite magnitude its order is not changed.

Cor. 2. Hence, if a function be developed in a series ascending by the powers of h, and h be diminished without limit, all the succeeding terms may be neglected when compared with any of the preceding.

45. There are intermediate orders of infinitesimals.

For let y2= ax, then r being taken an infinitesimal of the first order, y which is not finite cannot be of the first order, for then y or ax would be of the second, and is therefore of an intermediate order.

The student should consult the Scholium at the end of the first book of the Principia, where all this is exemplified by the different orders of contact in curves.

46. The product of two infinitesimals of the first order is an infinitesimal of the second order.

For let x and y be two infinitesimals of the same order, and take 1:x:: y: z, then s = xy; but when x is indefinitely small compared with 1, ≈ is indefinitely small compared with y, or is an infinitesimal of the second order.

Similarly it may be shown that the product of three is of the third order, &c. &c. .

Cor. Hence, if u = (x + h) (y + k)=xy + xk+ yh +hk,

VOL. I.

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