2

m.w to the first

w: and the first difference was formed, all through

The last terms of all the series were compared together, as a

π

check. Then for every term the arc less than was taken whose sine

π

4

or cosine (with proper sign) represented cos (wm.w). The natural

2

numbers were taken to the 7th decimal place, and differenced, as has been mentioned. The number of arguments thus computed is 5166; and the number of natural terms for the summation is the same; and the whole of these have been differenced on paper to the third order, and mentally to the fourth order.

The integration as far as w = 200 being thus completed, and with the utmost accuracy, the next step was to compute the integral from

w = 2.00 to w = infinity. Let u =

π

2

(w3 – m. w): the problem is now

where the last term may be integrated by parts as before. Proceeding

and so on, we find for the integral generally

(V6 − V1⁄2 + Vş − V。 + &c.) sin u + (V1 − V3 + V5 − V; + &c.) cos u.

6

The first limit of the integration being w, and the last being infinity, and the quantities vo, v, &c. vanishing for w = integral between these limits is,

infinity, the value of the

(− Vo + V2 − V1 + v. - &c.) sin u + ( − v1 + V3 − V5 + v, - &c.) cos u.

Making w = 2·00, computing these expressions for every value of m, and substituting them in the expression for the integral from w = 2:00 to w = infinity, the numerical value of the integral for each value of m was found.

This process was exceedingly accurate for all the negative values of m, and for the positive values about as far as m+30, when a difficulty presented itself. It must be remarked that when 3w - m (or in the

present instance 12 - m,) amounts to several integers, the values of the successive terms decrease at first with great rapidity: yet, in all cases, they increase at some part or other in hyper-geometrical proportion, and finally become greater than any assignable quantity. quantity. This will be seen most readily on observing the law of the terms when m = 0.

It is evident that these terms, however small may be the quantity 2 will at some stage receive in succession new multipliers, greater

3π.w3
2

than ; that after this they will increase, and that the rapidity of their proportionate increase will go on continually increasing. From this point then the magnitude of the terms will increase hyper-geometrically.

The value of the integral however will be finite; and a limit for the value remaining after the computation of any number of terms in the series v。, v1, &c. may be found. For, wherever we stop, the residual dv dvn term will be of the form cos u. or f sin u. : where is the dw

dv

;

dw

va

term last found in the series. Now it is evident that either of these quantities is less than dw for the magnitudes of the quantities to be integrated are always smaller, except in the particular cases when cos u or sin u = ± 1; and their signs are constantly varying as the value of w varies whereas the sign of is always the same. The residual in

dv
dw

tegral, therefore, is certainly less than v, the last term found in the series, and is probably much less: and therefore, if the last term computed consist only of integers in the last place of decimals which we

wish to retain, even though the divergence of the series be just beginning, the use of these terms will give the integral required with the utmost practical accuracy.

Now when m = +30 nearly, it is found that the divergence commences at v, or before it; and that the term v, is not so small that a quantity which is likely to be a sensible portion of it can be safely neglected. To approximate here, I have used the following consideration. It is known that the slowly converging series

A-B+C - D + E - F + &c.

may be converted into a series of the same kind with much smaller terms by putting it in this form,

14

+ } ( A − B) − ↓ (B − C) + ↓ ( C − D) − ↓ (D − E) + ↓ ( E - F) − &c.

In the instance before us, such a series would be produced by commencing the integration by parts with (v。 – v2) instead of v; and the residual term will be of the form of

dv. dw

dvn+2 dw

then the quantity dw

d

Now if the progression is stopped at such a point that is, for all

the following values of w, greater than dw (vn -vn+2) has always the same sign, and the reasoning above shews that the residual integral will be less than (n − Vn+2). A close approximation therefore

will be obtained by summing the series as if we had begun with (v。 - v2) instead of v. and this, it is easily seen, will be effected by taking half of the last term in each of the series multiplying sin u and cos u. multipliers which I have used are, in fact,

The

The doubt which remains extends, I apprehend, to digits in the fifth place of decimals, but no higher.

The number of terms computed by logarithmic process for this part of the integral is about 900.

I could have wished to extend the computation as far as m=+6·0, so as to include perhaps one or two more maxima of the values of intensity. Indeed, if it had been possible to foresee the approximate places &c. of maxima, I should probably have commenced the computations to that extent. The trouble however would be great, as the summation must be extended as far as w = 2.5. Should any person be disposed to go to that extent, I would recommend that the summation of the values computed in this Paper should be also extended, for the values of m beginning with +3.0.

I subjoin a Table of the different sections of the summation and remaining integration for all the values used in this Paper.