the "flow and plunge" structure is the rule. If it can be said (as I do not think it can) that no deposits similar to the loss are now in process of formation by lakes: neither have we at this time, any example of the accumulation of such deposits through the agency invoked by Richthofen, on any considerable scale; although the postulated conditions exist in not a few regions. All dunes and drifted desert sands show winddrift structure, as a necessary consequence of the varying velocity of the wind; and it seems to me that even in the presence of the supposed steppe vegetation, a condition of things under which that structure should nowhere appear, or should have been destroyed afterwards, is much more difficult to imagine than that, under the anomalous conditions of the "Champlain" period of depression, such conditions of aqueous deposition as we now find only exceptionally, should have prevailed more generally and for a longer time; a time, however, immeasurably shorter than that to which we must stretch our imagination for the formation of a thousand feet of dust deposit, brought by a wind so uniform in its direction and velocity as to leave no trace of the proverbial variability of that agent. And when we find ourselves driven to the supposition that this extraordinary wind did, moreover, drop its uniformly fine dust into the trough of the Lower Mississippi, leaving all the adjoining upland without a vestige for hundreds of miles on either side: the sum-total of anomalous conditions required to sustain the æolian hypothesis partakes strongly of the marvellous.

ART. XVIII.-On a method of swinging Pendulums for the determination of Gravity, proposed by M. Faye; by C. S. PEIRCE. [Read before the National Academy Academy of Sciences, April 17th, 1879, with authority of the Superintendent of the U. S. Coast and Geodetic Survey.] AT the Stuttgart, 1878, meeting of the International Geodetic Association, M. Faye suggested a method of avoiding the flexure of a pendulum-support which promises important advantages. The proposal was that two similar pendulums should be oscillated on the same support with equal amplitudes and opposite phases. If the pendulums could be made precisely alike, the amplitudes precisely equal, and the phases precisely opposite, it is obvious that the support would be continually solicited by two equal and opposite forces and would undergo no horizontal flexure, except from the distortion of the parts between the two edges. But since none of these three elements can be made equal, it is necessary to inquire what would be the effect of such slight imperfections in their equalization as would have to be expected in practice.

I had the advantage many years ago of learning the main characteristics of the mutual influence of pendulums from Professor Benjamin Peirce. As my father's studies of the subject were never, I believe, written out, I am unable to say definitely what I derive from that source. But the truth is the little knowledge I have of mathematics was learned from him, and from him I got a clear idea of the nature of this particular problem; so that acknowledgments of detail, even if I were able to make them, would be quite inadequate.

In M. Faye's proposed experiment, four finite forces would be in operation, namely: the weights of the two pendulums, the elastic force tending to restore the two knife-edge supports to their position of equilibrium when they are both displaced together, and the elastic force tending to restore them when their relative positions are displaced. The system has, also, four degrees of freedom corresponding to motions against each of the four finite forces. Accordingly there will be four differential equations of motion. By neglecting the terms of the second order, these equations are made linear, and by the general theory of such equations, they indicate that each of the four motions of the system (viz., those of the pendulums and of the two knife-edges) is compounded of four simple harmonic motions. Two of these will have periods nearly equal to those of the pendulums; the other two will be mere tremors having periods nearly those of the natural elastic oscillations of the supports. These tremors will be so small that they may be neglected. In fact, if we simply suppose that the knife-edges are constantly in equilibrium under the various forces which solicit them (which is simply to neglect their living forces under their very small velocities) the tremors disappear, to the great simplification of the formulæ.

Putting, then, 4, and , for the momentary angles of displacement of the two pendulums, s, and s, for the momentary horizontal displacements of the two knife-edges, 1, and 7, for the lengths of the two equivalent simple pendulums (on an absolutely rigid support), g for the acceleration of gravity, and t for the time, we have

These equations are exactly like what we have in the case of a single pendulum on a flexible support; and I have shown their correctness in my paper on that subject.

There would be no difficulty in making the two pendulums so nearly alike that they might be regarded as entirely so in their actions on the stand, the whole amount of which is small. AM. JOUR. SCI.-THIRD SERIES, VOL. XVIII.-No. 104, AUGUST, 1879.

We may also consider the parts of the stand on which the two knives rest as equally elastic. We may therefore take (s,+s) as proportional to (+92), and (s,—s,) as proportional to (9,-2). Denoting, then, by x and y two constants whose values will be easily determinable by experiments we have

or

8, +8,=(x+y) (P,+93)

8-8,-(2-y) (P,- P2);

1

1

Substituting these values of s, and s, in the differential equations, and also writing l+dl for l, and l-dl for l2, they become

dP
(1+x+81) +y
dt

dt2

The solution of these equations is (A, B, t,, and t, being the arbitrary constants)

g

- }

(t-t2)

{1 + x + √(8)2 + y2

g

The condition that the pendulums are started by drawing them away from their positions of equilibrium and then letting them escape nearly at the same instant makes t, and t, nearly equal. We may reckon the time from the mean instant of starting. Then at that instant we have very nearly

9,=−A (2+√1+22) − B (2−√√√/1+%3).

And since the amplitudes are nearly equal and the phases nearly opposite,

or A+B (nearly) A (2+1+2)+B (2−√√/1+22)

1.

There would be no insuperable difficulty in making the pendulums so near alike that dl should be less than y, even if the latter quantity were smaller than it would be likely to be. But it will be seen presently that care must be taken in the construction not to make y too small. We shall have then dy or z<1; whence B<A. Thus the amplitudes of the first terms in the expressions for both, and y, are greater than those of the second terms, while the period of the first terms is shorter than that of the second terms. From this it can be shown to follow that the whole oscillations of the two pendulums have the same period, which is that of the harmonic motions represented by the first terms of their values. Thus, in the figure, the abscissas representing the time, we have a wave of short period and large amplitude placed in comparison with a wave of long period and small amplitude.

F O

The phase of the short wave advances on the long one and goes over and over it. In each complete cycle of the curve representing the short wave, beginning and ending at y=0, it must cut the other curve twice unless the latter has mean time crossed the axis of abscissas once and not twice. When this happens there will be three intersections or only one, according to the direction of the crossing. Hence when the short curve has advanced over any even number of crossings by the long one of the axis of abscissas, the mean number of intersections per cycle of the short curve will be exactly two. Now let the short curve represent the first term in the expressions for y, or 2 and let the long curve represent the second term with its sign changed; then, the intersections will represent passages of the pendulum over the vertical, and it will be seen that there are two for each complete period of the quicker harmonic component of the motion.

2

The mean period, then, of the oscillation of either pendulum will be

Now let us suppose that dl is so small that

lected, being less than one millionth. This would happen, for instance, if 7 were one meter, y a half a millimeter (so that the stand would be somewhat less stiff than the Repsold tripod), and ôl were one twenty-fifth of a millimeter, so that the difference between the natural times of oscillation of the two pen

dulums was not over four seconds a day, a perfectly attainable adjustment. Then the period would reduce to

The terms x-y here indicate that the apparatus would still be subject to a correction for flexure: but it would be only for the relative flexure due to the distortion of the support between the two knife-edges. This could of course be made very small. It would still have to be measured: but it would be measured once for all, since it would be the same at all stations. At present, the measurement of the flexure at each station, involving as it does the erection of a separate pier, threatens to be one of the most troublesome and expensive parts of the whole work of determining gravity. This would be entirely obviated by M. Faye's plan, except that the small differential flexibility would have to be determined once for all. The proper way to make the stand so as to bind the two knives to their relative position as firmly as possible while allowing a moderately large flexibility to the whole stand, so that the two pendulums could freely influence one another, would easily be found out.

2.

WWW

The accompanying figure, for instance, represents one such arrangement as viewed from above. T, T, are tongues upon which the pendulums would rest. These would be cast in one piece with the heavy frame F, F, F, F. This frame would rest on four legs L, L, L, L, which would spread at the bottom in the direction of the motion of the pendulum. At the bottom they would be bolted into another heavy frame. The cross braces C, C, C, C, would prevent twisting.

The average period of oscillation of either pendulum, after correction for flexure, would be that belonging to a simple pendulum having the length 1, the mean of the lengths of the two simple pendulums whose natural periods of oscillation. would be the same as those of the given pendulums. But although this would be the average time of oscillation of either pendulum, yet neither pendulum would have all its oscillations of the same duration. It is, therefore, necessary to inquire what error might arise owing to the observations not extending over any exact number of cycles of motion, so that the mean