until the desired part of the spectral field is brought into the observing eyepiece. If the spectrum is now either too high or too low, it shows that the refracting edge of the prism is slightly inclined to the mirror-face, and the screw c is turned until the spectrum is centred. Then, if all the preliminary adjustments have been properly made, the angular deviation of the central ray in the field will be given by the relation 0=2(B― (90°+a)),

where is the circle-reading for a deviation 0, and a is the zero-reading determined as already described.

In very accurate spectrometric work it is important to determine just what degree of accuracy is required in making the various adjustments of parts to each other in order to attain a given degree of accuracy in the final result. The theory of these adjustments is comparatively simple, but somewhat lengthy, and it will therefore be briefly indicated in a future paper.

Astro-Physical Observatory, Washington, D.C, March 1893.

IN

XXXIX. On the Highest Wave of Permanent Type. By J. McCOWAN, M.A., D.Sc., University College, Dundee*. N a previous communication †, in which I discussed the general theory of the class of waves in water or other liquid which have no finite wave-length but which are of permanent type, that is to say, which are propagated with constant velocity without change of any kind, I gave a rough estimate of the maximum height to which such waves might attain without breaking. The paper dealt chiefly with an approximation which was specially suitable for waves of small or moderate elevation, and it is the object of the present paper, therefore, to supplement this by investigating an approximation better adapted to the discussion of the extreme case of the wave at the breaking height, and sufficiently exact for ordinary purposes. I trust, however, to be soon able to communicate a fuller discussion of the general theory of the solitary wave which I have almost completed.

1. The General Equation of the Motion.

The highest wave which can be propagated without change in water of any given depth is obviously the highest solitary *Communicated by the Author, having been read before the Edinburgh Mathematical Society, June 8, 1894.

+"On the Solitary Wave," Phil. Mag. July 1891.

wave for such depth; for the height to which waves can attain without breaking must evidently increase with their length, and the solitary wave may be regarded as the limiting type to which each individual wave, reckoned from trough to trough, in a permanent train of finite waves approaches as the wave-length indefinitely increases. In fact this paper and the former, "On the Solitary Wave," may be regarded as giving a very close approximation to the form and motion of the individual waves in a train of finite waves if the wavelength is even so small a multiple of the depth as ten or twelve.

It will thus be convenient to follow to some extent the methods and notation of the paper "On the Solitary Wave," and references to it will be briefly indicated by an S prefixed.

Consider, then, a solitary wave propagated with uniform velocity U along the direction in which a increases in an endless straight channel of uniform rectangular cross section, the axis of a being taken along the bottom and that of 2 vertically upwards.

Let the motion be regarded as reduced to steady motion by having superposed on it a velocity equal and opposite to the velocity of propagation of the wave, and take a=0 and z=c as the coordinates of the crest. Let u and w be the horizontal and vertical components respectively of the resultant velocity q in the steady motion at x, z, of which, further, is the velocity potential and the current function.

We shall now, referring to the "General Theory of the Wave," (S. § 1), seek to determine a form of the relation between +p, or, as we shall here find more convenient, u+w and z+a corresponding to S. (1) and (2), but only containing so many disposable constants as will suffice for the degree of accuracy at present desired. Noting that for the limiting form the velocity at the crest must vanish, and remembering Sir George Stokes's expression * for the leading term in the velocity near the crest of a wave at the breaking-limit, we shall assume (compare S. (6)):—

u+ww=−U{1—ƒk2 sec2¿m (z+ix) } √1− k2 sec2 §m (z+ɩx), (1)

where

k= cos mc ;

(2)

as a form conveniently integrable with respect to z+ux so as to give + in finite terms if so desired.

It should be noted that all the conditions required to be *"On the Theory of Oscillatory Waves," Appendix B. 'Collected Papers,' vol. i.

satisfied for a solitary wave, with the exception of the condition of constant surface-pressure, are identically satisfied by (1): for w vanishes with 2, and when +∞, u=-U and w=0. We proceed, therefore, to determine the surfacepressure with a view to the determination of the available constants so as to satisfy as nearly as may be this condition for a free surface: as it is, (1) may be regarded as giving a particular forced wave.

2. The Surface-Pressure near the Mean Level. Expanding (1), and writing for brevity

we get

a=1+2ƒ, B=1-4ƒ, &c.,

— (u+ww)/U=1−ak2 sec2 1m(z+ix) — } ẞk1 sec1 Įm(z+ix); (4) therefore, further, for a positive

— (u + ıw)/U=1—2ak2e—m(x−ız) +2(2a—ßk2)k2e—2m(x−12) &c.; (5) therefore, integrating with respect to z+x,

−m(y+18)/U=m(z+ix) +2ıak2e—m(x—ız) — 1(2a —ẞk2)k2ɛ−2m(x−iz). (6)

Now if h be the mean depth, or the depth at an infinite distance from the crest, we must, taking =0 at the bottom, have Uh at the surface. Also, if n denote the elevation of the surface at any point above the mean level, we must have z=h+n at the surface. Substituting these values in the expression for involved in (6), we obtain as the equation to the surface,

mn=2ak3e-m2 sin m (h+n) — (2a-Bk2) k2e-2m sin 2m (h+n).......; or, when ŋ is small,

η

mn=2ak2e-m* sinmh-(2a-ẞk2-2a3k2) k2e-2m2 sin 2mh...; (7) which gives the equation to the surface in a convenient form for points not too near the crest.

Again, from (5) we get

q2/U2=1—4ak3e-mx cos mz+4k2e-2m2 { (2a-ẞk2) cos 2mz+4a2k2}; (8) therefore, by (7), we have at the surface where n is small, q2/U2=1-4ak2e-m* cos mh

η

....

+4k2e−2mx { (2a—ẞk2) cos 2mh+a2k2(1+2 sin2 mh)} ........ (9) Now in a liquid of density p moving irrotationally acted on by no force but gravity, the pressure p at any point is given by

p=constant-pq2-gpz;

Phil. Mag. S. 5. Vol. 38. No. 233. Oct. 1894.

2 B

(10)

and therefore if Sp denote the excess of pressure at any point on the surface over that at the mean level, we have

(11)

whence, on substituting for ŋ and q2 from (7) and (9), we get Sp/pU2=2ak2e-mx{cos mh-g/mU2. sin mh} + &c. (12) Now for a free surface dp ought to vanish; therefore for a first approximation we must take

that is,

cos mh-g/mU2. sin mh=0 ;

U2=g/m tan mh ;

and (12) becomes, writing it out to the next term,

(2a— ßk2)

Sp/pU2=2k2e-2mx {(2a-Bk2) sin2 mh-3a2k2}.

The coefficient of e-2mr in (14) ought of course to be made to vanish; but it will be preferable for our present purpose to retain it as a small pressure-error, and so leave another of our constants available to satisfy the conditions in the neighbourhood of the crest to which we proceed.

3. The Surface-Pressure near the Crest.

Put z=c-, so that vanishes at the crest; then, writing for brevity

p =
= tan } me,
A=1-ƒ,

B={(1+11p2)ƒ— (1+3p2)}/8p,

we get, on expanding (1) in powers of 【―ux,

(15)

−(u+wwj/U=√ pm(?—ix) {A+Bm(¿—ix) + &c.}; (16) whence, integrating with respect to 3—ɩa, we get

m(†+18)/U—m¥√/U=2√p{m(5—ix)}3 {} A+}Bm(¿—ix)}. (1'

where is the value of at the crest.

then (18) gives for the surface, determined by

A cos + B mr cos + &c.=0.

π

Thus when r=0,=, showing that the crest is formed

by two branches equally inclined to the bottom cutting at an angle of 120°.

π

= τσ;
3

.. (19) gives when σ is small,

σ÷-Bmr/5A,

(20)

as a convenient approximation for the form of the surface in the neighbourhood of the crest.

Again, (11) may be written

whence, since must have

q and vanish together, to make Sp vanish we

(25)

q2/U2pmr {A2+2ABm}+&c.},

Sp=pr{g cos ¿mA2μU2+ABU2m}+&c.}.

Hence, since =+σ we must have, to make dp vanish to

3

This cannot vanish unless B vanishes, in which case we see by (20) that the curvature of the surface vanishes close to the crest. This result is obviously independent of our approximation, but we have not taken enough of constants to secure it here: it will be found, however (v. § 5), that the other equations determining the constants we have at our disposal will make B very approximately vanish.

4. Numerical Determination of the Constants.

There is still one important condition to be satisfied. To ensure the connexion between our separate treatment of the neighbourhoods of the crest and mean level, we must secure that the stream-line -Uh, bounding the distant surface, shall pass through the crest: or, in other words, the flow across any infinitely distant section must be equal to the flow