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the intervals into which w is divided. And as the same formula applies to every one of the intervals, it applies to the sum of all.
And the sum of all the partial integrals through each section is the whole integral through each section. Thus we find,
sum of computed values of cos 3 (w" – m. w) Integral through each o |
I - = interval x | as sum of corresponding 2d differences
17 -oT 5760 sum of corresponding 4th differences
sum of computed values of cos #(w-m, w) |
- I [1st difference following the last term — 1st difference = interval x
preceding the first term
+ 24 | 17 so difference following the last term – 3d difference | T 5760 preceding the first term }
This process was used throughout. For the purpose of forming the 3rd difference following the last term and that preceding the first term,
it was necessary to compute two values of cos #(w-m.w) following the
end of each section and two preceding its beginning.
The values of #(w – m. w) were computed by means of Delambre's
Tables Trigonométriques Décimales. The centesimal division of the circle is, in every instance in which I have used it, far more convenient than the sexagesimal: but in an instance like the present, where there is continual addition or subtraction of arcs, and where the whole arc amounts to several circumferences, the labour and liability to error would be so great with the sexagesimal division, as to make the operation almost impracticable. The numbers were thus formed in each section. The first
four values of : w" were computed independently and differenced, and with these differences the rest of the series of t w was formed: the
last was also computed independently as a check. The first term of
each series of ; (w” – m. w) was formed by applying — : m. w to the first term of the series of ; w; and the first difference was formed, all through
the series, by applying — s m . ow to the corresponding 1st difference of
2 check. Then for every term the arc less than i was taken whose sine
w". The last terms of all the series were compared together, as a
or cosine (with proper sign) represented cos #(w- m . wy. The natural
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 = 2:00 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 = #(w — m. wo): the problem is now
where the last term may be integrated by parts as before. Proceeding with this operation, and putting
and so on, we find for the integral generally
The first limit of the integration being w, and the last being infinity, and the quantities v, v, &c. vanishing for w = infinity, the value of the integral between these limits is,
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 = + 3-0, when a difficulty presented itself. It must be remarked that when 3 wo – 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. 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 s:- will at some stage receive in succession new multipliers, greater 7r. 3 T. wo - --- - - - - than 2 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, v, &c. may be found. For, wherever we stop, the residual
term last found in the series. Now it is evident that either of these quantities is less than s ; for the magnitudes of the quantities to be in
tegrated 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 - - dv, . ro- - w varies: whereas the sign of o is always the same. The residual integral, 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
may be converted into a series of the same kind with much smaller terms by putting it in this form,
In the instance before us, such a series would be produced by commencing the integration by parts with , (vo – v.) instead of v,; and the residual term will be of the form of
dv. is, for all
Now if the progression is stopped at such a point that dw
the following values of w, greater than *::::, then the quantity # (c.-w, ...)
has always the same sign, and the reasoning above shews that the residual integral will be less than , (v. - v, , ). A close approximation therefore will be obtained by summing the series as if we had begun with , (v,-v.) 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. The multipliers which I have used are, in fact,
– v. 4. v.-v, + A vs............for sin u,
– v. 4 vs – v. 4. A v............. for cos u.
The doubt which remains extends, I apprehend, to digits in the fifth place of decimals, but no higher.