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to this definition immediately offers itself, viz. that if lim.-. {f(x)} is

an infinitesimal of the mth order, the function

f(x)

xmi

increases or decreases

indefinitely, while a converges indefinitely towards zero, according as m1

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f(x)

xm

converges towards a finite limit while a converges indefinitely

f(x)

Xm1

and as towards zero, it depends upon the sign of m, -m whether increases or decreases indefinitely at the same time. But if no finite and positive value of m can be found which will render lim a finite quan

S⋅f (x)\

x=0

xm

tity, there are two cases to be considered. 1st. If f(x)

f(x)

0

increases indefinitely towards

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converges inde

finitely towards zero along with a however great m may be taken, it follows from the general definition of an infinitesimal of the mth order that lim.= {f(x)} is an infinitesimal of an infinitely high order. 2d. If 10 for values of x which converge indefinitely towards zero, however small m may be taken, it follows from the same general definition that lim {f(x)} is an infinitesimal of an infinitely low order.

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2.

theorem.

x = 0

Of infinitesimals in general I may enunciate the following

x = 0

THEOREM.

If lim {f(x)} is an infinitesimal of the mth order, and if lim {(x)} is an infinitesimal of the mth order, the equation ƒ (x) = q(x)

x=0

cannot exist for the general value of x.

DEM. For if f(x) can be equal to (x) for the general values of x, dividing by a", we find that

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towards a finite limit as a converges indefinitely towards zero, whilst

1 xm-m2

1 0

converges towards or zero at the same time, according as m

is greater or less than m. Therefore a quantity converging towards a finite limit can throughout be equal to another which converges

1

towards or zero, which is absurd. Therefore f(x) cannot be equal to

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COR. If lim x=0 {f(x)} is an infinitesimal of the mth order, and lim.=0 {(x)}, lim.= 0 {1 (x)}, lim,-. {p. (x)}, lim .-. {Pm-1(x)} infinitesimals of the mih, mh, m- orders, f(x) cannot be equal to

th

= 0

Ap(x)+Bp,(x) + Cp2(x)

for the general value of x.

SCHOLIUM.

Hence we see that the law of homogeneity, which is so essential an element of all analytical developements, holds good at the limits of the functions and variables as well as for values varying between finite limits. We shall now see that this law is alone sufficient to demonstrate La Grange's principle within the proper limits of its application, as well as to indicate at once the cases in which it is necessarily inapplicable.

3. I shall now enunciate and demonstrate La Grange's principle.

THEOREM.

If f(x) be a function of x continuous between the limits 0 and 1, and if limo{f(x)} is an infinitesimal of a finite and positive order ∞, the function f(x) for any value of x within those limits may be analytically represented by a series of terms of the forms Ax" + Bx2 + Сx2+&c. where A, B, C are finite coefficients, and a, ß, y finite and positive exponents.

DEM. As lim {ƒ(x)} is an infinitesimal of the finite and positive

x=0

order, it follows that the limit of the ratio

f(x)

xa

for the values of x

which converge indefinitely towards zero, is equal to a finite quantity. This finite quantity is the coefficient 4. From this and from the continuity of the functions (the difference of two continuous functions being itself a continuous function) it follows that if we make a increase insensibly from zero to some finite quantity within the limit a the values which the functions

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successively assume, may be respectively represented by Ba, Ca, Dæð where B, C, D are finite, and a, ß, y, & finite and positive with the law ẞ>a, y>ß, d>y. Therefore, reducing and transposing, we see that for any finite value of a within the limit x, f(x) may be analytically represented by the series Axа + Bxß3 + Cx + Dx© + &c. Q. E. D.

COR. We may from the preceding proposition deduce a mode of finding successively the terms Axa, Bx3, Ca, and thus of actually effecting the developement of f(x). This is best explained by an example. Let f(x) be sin a and assume sin x = x + Bx2 + Cx2+&c. Dividing sin a by a we get = A + Bxß-a + Сxr-α + &c. Now making a conxa verge indefinitely towards zero, as ẞ>a and y>a, it is manifest that where a is that finite and positive number

A is equal to lim x=0

(sin x)

which can render lim

x = 0

sin x xa

α

2) a finite quantity. But by the ordinary

rules of the Differential Calculus for finding the values of fractions which

0

for certain values of the variable become we find that

0'

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which for a=1 becomes finite and equal to unity. Therefore

sin x=x+Bx2 + Сx2 + &c.

treated sin x, we get ẞ-3 and B=

Treating the function sin x-x in the same manner as we have just

=

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This process is general, and may be easily applied to demonstrate the theorems of Taylor and Maclaurin.

4. The theorem of the last article is La Grange's principle, and was used by that analyst as the fundamental principle of his Calculus of Functions. By the corollary to the theorem in Article 2, it is clear that it is inapplicable to functions whose limits for the values of a which converge indefinitely to zero are infinitesimals of an infinitely high or an infinitely low order, Axa, Bx, Cx being at the same limit infinitesimals of finite orders. There are however only two known classes of functions which have this property, viz. ea and

1

1

log x'

The limit of

the ratio of eto a" is easily shown to be infinitely small, however great a may be taken. Lim. (e) is therefore an infinitesimal of an infinitely high order, and consequently e cannot be represented by a series like Ax + Bx2 + Cx2 + &c. On the contrary, the limit of the ratio may be shown to be infinitely great, however small a

1

of to a
log x

may be taken.
be taken. Lim

x=0

(log a

Lim = log

is therefore an infinitesimal of an infi

nitely low order, and therefore cannot be represented by a series such as Ax + Bx2 + Cx + &c. In either case indeed, if we assumed the principle, we should, by passing to the limits of the equivalent series, find an infinitesimal of an infinitely high or infinitely low order, equal to a series of infinitesimals of finite orders, which would violate the principle of homogeneity which exists equally in finite and infinitesimal quantities.

XIV. On the Transmission of Light in Crystallized Media. By PHILIP KELLAND, B.A. Fellow and Tutor of Queens' College, Cambridge.

[Read Feb. 13, 1837.]

INTRODUCTION.

THE object which I have principally had in view in the Memoirs which I have hitherto laid before this Society, has been the development of the equations for the motion of a series of particles in a form calculated to lead to a simple and tangible interpretation.

The point of greatest interest connected with the subject, is the determination of the law of force by which the particles act on each other. The data for the investigation of this law are neither numerous nor well defined, and one difficulty in particular attaches itself to every part of it, arising from our uncertainty respecting the number and nature of the causes which may conspire to the production of any particular phenomenon.

In my first Memoir I discarded all complexity from my investigations, and conceived the whole effect to be due to the action of particles of the same kind: from a comparison of my results with those of observation, I was led to the conclusion that the law of force is that of the inverse square of the distance, and by means of that law was enabled to shew that the vibrations are necessarily transversal.

In my second Memoir I treated the subject in a more general manner, attributing the phenomena to the action not of one system of particles, but of two, which act mutually on each other. There appeared numerous coincidences, which, if they did not suffice perfectly VOL. VI. PART II.

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