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9, p and J being the angles which a straight line makes with the co-ordinate axes of a, y, x. For simplicity put

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p being the projection on the line making angles 6, p, N, with the axes, of the distance OP of the point P from the origin: the above equations then become

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It is true a more general solution of these equations would have been obtained by assuming the values of k, 6, p, N, different for each of the three equations, but as the complete solution will consist of a series of terms similar in form to the above, it is sufficient for our purpose to exhibit that term above which has the same period for each of the three directions, and which consequently corresponds to one and the same undulation.

From these equations it is manifest that the same state of motion 2 or 7 a wave; also the motion at the end of t + At is the same at the point

recurs when kp is increased by 2nt, and consequently o is the length of

p + Ap as at the time t at p, when n At = Ap, whence the velocity of

- - d transmission parallel to p = # = ??.

It may be worth while to notice here that the proposition which we have considered assumes the velocity of transmission to be the same in all directions: in general, however, this will not be the case, the direction of transmission being defined by the simultaneous transmission of a system of waves, and the velocity will have reference to that direction only; but as % is independent of any particular direction, and depends only on the nature of the substance, it must be either the velocity of transmission itself, or the velocity in a direction making a constant angle with that of transmission, and consequently varies as that velocity. And for the same reason p must be either in the direction of transmission, or making a constant angle with it, and as the introduction of a constant cannot in any manner, affect our results, we

may consider p and # respectively as defining the place and velocity

of the wave at the end of the time t.

Another remark is also important, that since from the constitution of the medium it is indifferent in what direction the axes of co-ordinates are taken, all the functions which we may introduce involving 3p must finally turn out independent of 6, p and N, so that we might at once suppose the direction of transmission to be the axis of y, and put 3 y for 3p; this, however, I shall not do, as it does not appear necessary, and it is convenient to retain the symbol p, on other accounts to be noticed hereafter. The above remark will be mainly useful in pointing out to us what are the quantities to be rejected in our equations of motion.

Let us now take as the solution of equation (1) the form we have obtained from equation (2), which is perfectly allowable, since the latter is only a particular case of the former: the quantities n and k are of course not necessarily the same for both.

Put the solution under the form

a = a cos (ct kp); ... §a = a cos (ct—kp—köp) a cos (ct—kp) = a cos (ct—kp). (1—cos köp) + a sin (ct-kp) sin köp,

where 3p=3a cos 0+39 cos p + 82 cos V is the projection of the distance PQ on the line OP.

By the substitution of this expression for 8a, and analogous ones for 33 and 3-y; our equation (1) becomes

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now it is manifest that Xq(r) sin köp = 0; and also because sin (ct—kp) is independent of >, and the term > ro 3a, §y sin köp is of an odd order, wherever there is a positive term, there will be a corresponding negative one: the whole expression denoted by the symbol X in this

term will therefore be identically equal to zero.

Precisely the same reasoning applies to all similar expressions.

Further as regards the term > ry, 8a 3 y sino *p, bearing in mind

the remark above made with respect to p, it is manifest that the part

so sin” sy will have one and the same value, for two equal

values of 3a, with different signs: thus, to assist the conception refer

ring to the cube as at the commencement of this paper, the point P

being at its centre, suppose a particle at each of the two upper cor

ners of the face on which you are looking, and y vertical, then the F(r) sin” kò

expression s == *** is the same for each of them, but 8a in

one corresponds to -3a in the other, and the sum of the above func

tion for two such particles vanishes. It is clear, therefore, that this - F . . k.8 - expression sersy on sin" **, and all analogous ones are identically

equal to zero.

Our equations then become much reduced, and assume the remarkably simple form,

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F F *} = X {q}(r) + ro 3y"; - X {q (r) + £ose. Now it must be observed that we have not deduced the above equations directly from the equations of motion; but have obtained them by first solving for one particular case, and assuming that the same Jorm holds in the -general one: our solution is

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n°–2X {p(r) + ro 3a*}. These results appear to be very simple in their form, and recommend themselves from the readiness with which they can be applied. It is true, we have not obtained them on a general hypothesis, but I think we may venture to say they rest on one which carries with it an air of probability; and I confess there seems more difficulty to conceive an hypothesis different from this for uncrystallized media, than to concede this. It is, moreover, the same which M. Cauchy adopts, but the results obtained differ in one especial point, viz. that his assume, and are of so general a form, that little construction beyond the explanation of dispersion can be put upon them. Professor Powell has, it is true, deduced from them the expression H o for the velocity. I shall make no remark on this deduction, as it arose from the simple consideration of one attracting particle, which is too limited to be regarded as even an approximation to a general result. I shall merely observe that in the sequel, owing to the negative value of one of the terms (l) there adopted, it is clear from experiment that the above form is incorrect.

It is true, some subsequent hypothesis might be necessary to adapt the formula as we have it to all cases, but for the present we have a form as simple as possibly can be obtained, and whose interpretation will be a matter of little difficulty. Before, however, I proceed to such interpretation, it may be useful to examine how it applies to the known dispersions of a number of glasses and other substances, since, unless it has some pretensions to supply us with results coinciding with those of observation, it can have little claim on our notice.

SECTION II.
Examination and Illustration of the Formula.

LET A represent the length of a wave; v the velocity of transmission.

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or the same state recurs after intervals of #, which is consequently

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