In the last integral of all, we deform the contour into a circular arc of indefinitely great radius, starting and ending at. Since n is supposed to be positive the integrand is 0(t) on the deformed contour, and so the integral is zero. We thus obtain the formula

Now that the large variable n only occurs in a single term of the integrand, we proceed to apply the methods of Debye by choosing a contour on which

is purely real.

(t-1/t) - log t

Writing tree, we see that the contour has to satisfy the condition (r+1/r) sin 0-20=0;

this equation is satisfied if 0-0 or if

=

r=0 cosec 0{1±√(1-0-2 sin2 0}.

Taking the upper sign, so that

r=0 cosec 0{1+√(1-0-2 sin2 0)},

(3)

we obtain a contour of the required type if we take to vary from -π to π.

The contour, which is symmetrical with respect to the real axis, passes through (1, 0) and has an abrupt change of direction at that point; its direction immediately above the real axis is inclined to the positive direction of the real axis *.

where is given as a function of 0 by equation (3), t=rei

and t

*This is easily proved; it is suggested by the fact that (t-1/t)-logt has a triple zero at t=1. Various properties of the contour are given in the first of my papers to which reference is made in § 1. The curve obtained by giving the lower sign to the radical is the inverse of the curve obtained by taking the upper sign.

In my previous paper, it was convenient to call this function F(0, 1).

Sincer is an even function of 0, we easily deduce that

But the contour starts from (1, 0) in a direction making an angle with the line joining this point to the origin; {r π and so tan- sin 0/(1-r cos 0)} decreases from 3 to 0 as increases from 0 to 7, while tan-r sin 0/(1+r cos 0) increases from 0 to π in the same circumstances. We therefore write equation (4) in the form

2r sine F'(0)de, (5) ¤}

1-2

it being observed that F(0) = 0, F(T) = ∞, so that the integrated part vanishes at each limit.

Now F(0) is positive when 0≤ 0 ≤π, so that F(0) is a steadily increasing function of e. Moreover, we can show that -tan-12r sin 0/(1—r2)} is a steadily decreasing function of 0. For we have

Now + sin cos 0-20% cot

vanishes when 0=0 and has the positive derivate 2(cos 0-0 cosec )2; hence it is positive when 0 <0. The function

π-tan-1 {2r sin 0/(1-2)}

therefore has a negative derivate, and the desired result is proved.

Taking F(0) as a new variable, F, and writing (F) for the function-tan-1 {2r sin 0/(1-2)}, we have

L'J.(ne)dx = 4√ (F)edF,

П

0

where (F) is a positive decreasing function of F such that $(0)=π.

Hence, since the integral on the right is uniformly convergent for large values of n*,

Lim

1

1

: S

[n['Jn(ne)dx] = Lim S® (u/n)e="du

That is to say,

0

it is evident that the function on the left steadily increases (for all positive values of n) as n increases.

3. When n is an odd integer we can express the integral as a sum of Bessel functions, since we have

=

{J1(y) —2J,'(y) — 2J,'(y) — ... — 2J',-1(y)}dy

= 1−{J。(n)+2J,(n) + 2J2(n) + ... + 2J2-1(n)}.

* Bromwich, 'Infinite Series,' pp. 434, 436.

When n is an even integer, we find that

n( 'J1(nx)dx= - S. * Jo (y)dy − 2 { J1 (n) + Js(n) + ... + J2-1(2)}.

0

The integral on the right does not seem to be expressible by elementary functions, but we have

by integrating the ordinary asymptotic expansion of Jo(y). The following table indicates the mode of increase of

1

n("J„(næ)dæ to its limit (namely †) as n increases through

the odd integral values 1, 3, 5, 7, ... .

4. We can obtain a closer approximation to the integral

in the following manner :

When is small we have

0+ sin cos 0-202 cot 0805/45,

02 — sin2 0 01/3,

02- sin cos 0404/3,

F(0)∞ 403/(9/3).

Hence from (6) we find that π-tan-1 {2r sin 0/(1-r2)} is a function of approximately equal to

and the complete powers of only.

}π-02/(5√/3),

expansion of this function involves even

Since F() is an odd function of 0, it follows that (F) is expansible in a series of ascending powers of F3, convergent when F is sufficiently small, in which the first two terms are given by the formula

(F)π-23F3/20,

and hence the integral of Jn(na) possesses an asymptotic expansion in which the ratio of consecutive terms is of order n3, the first two terms being given by the formula

2331

20πη
23

This approximation gives the value of the integral correct to four places of decimals when n=23.

XLI. Notices respecting New Books.

The Electron. Its isolation and measurement and the determination of some of its properties. By ROBERT ANDREWS MILLIKAN. Pp. xii+268. The University of Chicago Press. Price 7s. net. TARTING with a brief historical account of the rise of the

START

electron theory, the author soon reaches the question of the determination of the electronic charge e, and, after describing briefly the early work of the Cavendish school, and the difficulties encountered, he devotes special attention to the experiments carried out by himself and his students in the Ryerson laboratory of the University of Chicago. In these experiments, in place of Wilson's cloud, a single oil drop was observed; and Professor Millikan gives a most interesting account of the method by which the capture of single electrons by a drop was observed, the corrections to Stokes' law necessitated by the small radii of some of the drops, and the final determination of e to be (4·774+·005) × 10-10 electrostatic units, and N, Avogadro's constant, to be (6·062+006) × 1023. He also details experiments carried out on the Brownian movement in gases to determine N.

Not long before the war Ehrenhaft published an account of series of experiments which he considered to demonstrate the existence of a sub-electron, or charge very much smaller than the electron. Few English physicists found the work convincing, but, nevertheless, it aroused some attention. Professor Millikan devotes a chapter of his book to discussing this question of a sub-electron, and brings forward very strong arguments, based on his own experience, for supposing Ehrenhaft's results to be a development of experimental errors and uncertainties. There seems no doubt that, according to the best experimental evidence at present available, the electronic charge e is constant and indivisible, as was universally assumed.