(1-4) sec X

l2 (secx-cos x) (1−2§2 cos 28+ §1) (1—w2¿2)

(1-4){cos (-24) — w22 cos (+2¥)}
(1-2w22 cos 4y+w11)

— w2§2
- 25 sin (28—4v) {sin (6—24) — w2♬ sin (p+2¥) } },(65)

(1-22cos (28-4¥)+§1)

an expression very similar to that for I/Io given in equation (49). Reverting now to the terms containing Y in (58), let us evaluate

The expression for y, is given in (56): multiplying up Yy=-A2cfw2e-2pit [cos (2nd-a-f)-cos (2p2t+a--2nd)]. Integrating from 2nt, to infinity, and writing for brevity

× [secx.cos (2nd-a-y)—cos {2n(p2t2—¥)+A}]. Summing this from n=1 to n=∞, and multiplying by two,

4].. (68)

cos (A+28-44) —2 cos ▲
1-22 cos (28-4¥) +§a

Calling this I", and dividing through by I, we get

This last expression is independent of the thickness of the slab

of electrolyte.

The total expression for

I。= b2(secx- cos x) (1-2 cos 28+§1) (1—w2¿2)

(1—§1) [cos (4—24) — w22 cos (p+2y)]

-2 sin (28-44) [sin (p-2)-w2sin (+24)] (1-2w22 cos 4y+w1§1) (1 − 2§2 cos (28—4¥) +§1) secx [cos (y+a—28) — §2 cos (y+a)]

cf. 2

-2

secx- cos x

+

{

b2[secx-cos (x-2)]

sec x-cos X

As a matter of arithmetic this expression is fairly quickly

calculable if

I

has been already worked out; a large number

of the constants, &c., are the same in the two cases. The curve (2) (fig. 3) has been calculated from the same data as It I,

were given on p. 326; the two curves and Io

may be thus compared. In the former the maxima were slightly shifted back, in the latter the minima are slightly shifted forward. The proportion of energy absorbed by the plate is

(71)

Fig. 4.

It+ Ir

{1-+ asymptotes towards the value 1—1"/Io; that is

to say, when the layer of electrolyte gets very thick all is absorbed but the ray reflected from the first surface.

putting a zero. Taking the three parts (65) (69) (70) separately, the first becomes at once

It

where is the expression given in (51) or the following

The third part gives simply 62. Adding up we get

I,

The expression for also becomes greatly simplified if we take the incident ray to be non-damped. In this case

If we now put a=0, we have a==b, and 8=pata, and so on, as given on p. 325, and the expression becomes

which is the ordinary expression for 'Newton's Rings by Reflexion.'

VI. Discussion of an Experimental Case and a Correction. This section is independent of the rest of the paper, being devoted to the discussion of a correction necessary in the experimental case mentioned at the beginning.

It has been stated already in the introduction that the theory, sections IV. & V., does not give numerical results agreeing closely with the experimental ones. For all the values of B2/B1 I tried the calculated points lay continually below the experimental determinations. Dr. Barton found exactly the same peculiarity in his analogous experiments, and suggested that it might be due to the coursing of the wave-trains backwards and forwards along the wire circuit between the oscillator and the closed end. Working from this idea he arrived at a correction formula*, which, however, is not immediately applicable to our case. The following analysis is equivalent to Dr. Barton's though differing in method.

Let ABC (fig. 5) represent our circuit, B being the electrolyte, A the oscillator, Ĉ the closed end.

Let r and d be the reflexion and transmission coefficients at the electrolyte, with regard to energy, so that

I=tIo, I=rI。.

Let e and x be similar coefficients for the two halves of the X

*Thesis,' p 16, section V.