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- - - - - Now on expanding the sine, it is clear that when o: becomes greater than unity, or which is the same thing, when 8a becomes greater

than 3. we must put the supplement instead of the arc in the expansion; but we saw reason in the case of light to suppose that the expression for the whole force has the same sign, as the expression for the force exerted by those particles only which lie within the range of the first half wave.

In fact, if the different half waves give different signs, it is evident that they must give them alternately; and thus the above hypothesis would be confirmed; we shall therefore retain only the first term in the

- . . kõa: - - - - - expansion of sin” T2 . and reduce our investigation to the consideration

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of the sign of a *******- 3a*, taken within the range of the first 7"

half wave.

For every value of 3 y within this range there is a corresponding equal value of 8a, and vice versd: so that we may write the above expression as follows;

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27-43

- Or n.(3y'+ 3a')–43/3a*

2 r"+3 - ;

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from which it appears that if n = 2 our expression is — a n

essentially negative quantity. This conclusion was seen to be requisite,

in order to satisfy the conditions of direct vibrations with repulsive

forces. It is also evident that the same conditions would be fulfilled by

making n greater then 2, whereas if n be zero or negative this will not be the case. The hypothesis, which makes n equal to unity, we

shall examine hereafter.

5. By integration of the equation (1) and reduction, we obtain for the square of the velocity of transmission,

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which, if e be put for the distance between two consecutive particles of A, and E for that between two of B, and we make

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:

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For the present, we will omit that part of the expression which depends on the length of a wave; hence we have

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6. In the same manner, denoting by accented letters the quantities corresponding to a, &c. for the motion of a particle of medium B, we obtain

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7. If we were to suppose every part of medium A to be mixed with a portion of medium B according to a given law, all that we should require would be the direct integration of these equations, considering M a constant quantity depending on the relative natures of the media: but it will be more analogous to the nature of the question when applied to air or ether, if we suppose a want of uniformity in the mixture. Conceive, for example, that a given mass of medium A, impregnated with medium B, is enclosed by other portions of the same medium not thus impregnated.

Let e, be the distance between two consecutive particles in the

latter mass: then it is easily seen, that the attraction of the mass of 2

A, in the mixture, on a particle at its confines is C P. the quantity

- e

C depending on the mass so impregnated,

and that of the mass of B is C

M.
E.

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hence the action on a particle of A is c{{ + ...), but this is retained at rest by a corresponding quantity of A at distances eo; Po M P2 c(#4 #) = Dž.

Similarly, the action on a particle of B at the confines of the medium gives us

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or the velocity with which the motion of a particle of B is transmitted, is equal to the same quantity for a particle of A.

8. Thus far we have neglected the influence, which the particles of matter with which the media are united, (such are the elements of solid bodies) exert directly on the motion. Such influence will be calculated by resuming our equations, and supposing, in addition to the forces exerted by the particles which have motion, other forces produced by particles at rest.

Let Aa, Ay, Az be the co-ordinates of a material particle, measured from the place of rest of the particle under consideration; Vol. VI. PART II. II

R its distance from that point: then it is manifest that the only difference in the form of this term will be, that in the present case the difference of the co-ordinates of the two particles parallel to the axis of a, instead of being Aa + a + 3a – a, is Aa — a ; whence we obtain, supposing the force to vary inversely as the square of the distance,

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and the velocity is independent of Q; the only effect which is produced by the material particles being an indirect one, arising from the alteration of distance which they produce in the particles themselves.

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