pulley, ring, or sliding knot, so as to pass easily from one part of the cord to the other, then the weight will find the lowest point to which the cord will allow it to descend, when it hangs in the centre. The two parts of the fixed cord will then be of equal lengths, sustain equal tensions, and form equal angles with the vertical direction of the line which supports the weight.

Now the force which stretches the fixed cord, represented in the figure below by the two weights W and V, may be conceived to produce two different effects, one tending to bring the pins, or hooks, to which it may be attached, nearer to each other, and the other to bring them downwards in obedience to the gravity of the weight. These two effects may be represented in amount by the respective lengths of a vertical and a horizontal line meeting at a point perpendicularly above the pulley to which the weight is attached. The figure annexed will render this intelligible.

measure.

P

If the lines CA and CB, be supposed to represent inflexible rods, and the points of attachment A and B, be capable of sliding along a rod, or wire, towards D, then the weight P remaining always the same, the two wires CA, CB, must sustain the same tension as when they were flexible cords; but the resistance to a horizontal motion in the direction AD, or BD, is what we have to The tension of CB produces two different effects, one in the direction DB, the other in that of DC; and the lengths of these two lines respectively express the intensities of the forces into which CB is decomposed. But as CB is always of the same length, whatever its inclination to AB, it may be taken as the radius of a circle, and then DB becomes the cosine of the angle CBD, but the sine of DCB, or of its supplement PCB; and DC is the sine of CBD, but the cosine of DCB, or of its supplement PCB. The effect of the tension of CB towards sustaining the weight P, is, therefore, expressed by the line CD, or the sine of the angle of depression of the wire CB. But the tension of CA produces another similar effect, and to the same amount; of the two, therefore, the sustaining force is expressed by the double of CD, or

twice the sine of the angle of depression. This is represented by DE, the diagonal of the parallelogram AEBC. If we suppose the points of attachment, A and B, to be prevented from approaching each other by cords passing over pullies at some distance from these points respectively, but in the direction of AB produced, the weights which would be required to retain the points in their present positions, would be, to the weight suspended at C, as the length of AD, and of BD, respectively, to the length of DC. But these two lines have to each other the relation of the cosine to the sine of the angle of depression, and these two quantities will vary with every change of the angle, so that when one of the forces is constant, it becomes necessary for convenience in practice, to substitute for the relation of cosine to sine, its equal, viz. that of radius to the tangent of the angle of depression.

We may readily perceive that when the two wires are brought into a horizontal position, as there is no angle of depression, there can, of course, be no tangent, or the tangent will be represented by zero; whence the weights which draw the points A and B apart, must be to the weight which is suspended at the centre, as a finite quantity to one infinitely small, or nothing. Or if the weight hung upon the centre be of finite magnitude, the forces which draw the points A and B apart, must be infinitely great.

When, on the contrary, the two wires, AD, BD, come to a vertical position, the angle of depression is 90°, whose tangent is infinite. The relation, therefore, between the radius and tangent, is, in this case, indefinitely great. The whole weight is sustained by the horizontal rod to which the others are attached, and no force, however small, can be applied in a horizontal direction, tending to separate the upper ends of the wires, without actually drawing them apart, provided they move without friction, or other modifying cause. Thus we find the two extremes of this modification of the apparatus.

Let us now suppose the whole to be reversed, and the traction to be converted into pressure. This will give us the same conditions, and furnish an opportunity of testing, by actual experiment, the truth of the principles above examined.

For this purpose, on the horizontal plane in the middle of the frame, as represented in the following figure, are two planes, S, S, sliding in grooves, which confine them in the lateral directions, and furnished beneath, each with four friction rollers, on which they move with great facility in the longitudinal direction. The supporting plane is furnished with two pullies, over which cords pass, along a groove in the same, until they are finally attached to hooks at the two exterior ends of the sliding planes, S, S. When the weights R, R, are attached to the lower ends of these cords, they exert their whole force in urging the sliding planes towards each other. m, m, are two moveable planes, united together by a hinge, and each likewise connected with one of the sliding planes by a hinge, so that the four planes may either form one continued horizontal plane, or that the two moveable ones may rise to any angle with the horizon,

and be retained there by the weight P, suspended at the pin of the middle hinge, or may even come to a vertical position, in which latter case their under sides, or faces, rise together and come into immediate contact. The only difference between the case of the two wires before supposed, acted upon by traction, and of the two moveable planes now presented, acted upon by pressure, is, that in the former case, every equilibrium is stable, while in the latter, it is unstable; that is, when once destroyed, instead of returning to the position of equilibrium, the planes are urged with a constantly increasing force towards one or the other of the extreme positions of the apparatus. But this circumstance only serves to render more striking the actual equilibrium which may be produced, and in practice, no difficulty whatever is found in sustaining the parts of the

machine in the positions required. A weight equal to one-half that of the two moveable planes, is suspended by a small cord passing over a pulley, in order to deprive those planes of the effect of gravity of parts, which would otherwise materially influence the conditions of equilibrium. This counterpoise is seen at C.

In order accurately to measure the angle of elevation, a quadrant

(2) whose radius is of the same length as that of one of the moveable planes, is attached to the platform, and may slide along its edge horizontally, so as in every situation to adjust its centre precisely to the pivot of the lower hinge of that plane to which it is applied. The pivot of the middle hinge will then correspond to some point on the arc, and if the latter be divided into degrees and parts, the requisite data are immediately obtained, and the calculation is made with the utmost facility, by means of a table of tangents. Or a tangent rod T, as represented in the figure, may be added, having, on one side, lines drawn through the several divisions of the quadrant as far as convenient, and on the other, a scale of equal parts. The force P, then, is to one of the forces, R, as twice the tangent of the angle of elevation is to radius.

The machine which I have employed for this purpose, and which is represented in the figure, is about four feet in length by three feet and a half high; the moveable planes are about 10 inches in length, by 12 in breadth, and the sliding planes 12 inches in length by 24 in width.

The earliest account which I have seen of any mechanical implement constructed on the principle presented by this apparatus, is one contained in an old work by Ven Mandy, published near the beginning of the last century, where it is called an "augmenting power or force," and is merely described, without any investigation of its principle of action. It is there presented as a device adopted by a baker to facilitate the process of kneading his dough. It has of late years been applied to a variety of useful purposes, such as the construction of oil and printing presses, the cutting and perforating of metals, and other operations in which intense action through a very limited space is required to be produced by means of a small moving force.

Different names have been proposed for this kind of mechanical power, but none has yet obtained, I believe, a general adoption.

The term "toggle joint," is neither good English, nor in any way appropriate to the subject. The name of "progressive lever" has sometimes been applied to it; but from what has preceded, it will be seen that it has nothing in common with the ordinary lever, and though the latter may sometimes be employed as the medium through which the moving force is applied, yet it constitutes no essential part of the machine, any more than if applied to the middle of a cord to increase the tension in the rope machine. In reference to the mode of uniting the planes by hinges, or other equivalent means, I have proposed the term cardinary power, and to the particular apparatus above described, have given the name of tricardo.

On the Influence of the Air in determining the Crystallization of Saline Solutions. By THOMAS GRAHAM, ESQ. A. M. F. R. S. E.* THE phenomenon referred to has long been known, and popularly exhibited in the case of Glauber's salt, without any adequate ex

From the Transactions of the Royal Society of Edinburgh; but revised by the Author for the Phil. Mag. and Annals.

planation. A phial, or flask, is filled with a boiling saturated solution of sulphate of soda or Glauber's salt, and its mouth immediately stopped by a cork, or a piece of bladder is tied tightly over it, while still hot. The solution, thus protected from the atmosphere, generally cools without crystallizing, although it contains a great excess of salt, and continues entirely liquid for hours, and even days. But upon withdrawing the stopper, or puncturing the bladder, and admitting air to the solution, it is immediately resolved into a spongy crystalline mass, with the evolution of much heat. The crystallization was attributed to the pressure of the atmosphere suddenly admitted, till it was shown that the same phenomenon occurred, when air was admitted to a solution already subject to the atmospheric pressure. Recourse was likewise had to the supposed agency of solid particles floating in the air, and brought by means of it into contact with the solution; or it was supposed that the contact of gaseous molecules themselves might determine crystallization, as well as solid particles. But although the phenomenon has been the subject of much speculation among chemists, it is generally allowed that no satisfactory explanation of it has yet been proposed.

In experimenting upon this subject, it was found that hot concentrated solutions, in phials, or other receivers, might be inverted over mercury in the pneumatic trough, and still remain liquid on cooling; and thus the causes which determine crystallization, were more readily examined. For this purpose, it was absolutely necessary that the mercury in the trough should be previously heated to 110° or 120°; for otherwise, that part of the solution in contact with the mercury cooled so rapidly, as to determine crystallization in the lower part of the receiver, long before the upper part had fallen to the temperature of the atmosphere. In such cases, crystallization beginning on the surface of the mercury, advanced slowly and regularly, through the solution. Above, there always remained a portion of the solution too weak to crystallize, being impoverished by the dense formation of crystals below. It was also necessary to clean the lower and external part of the receivers, when placed in the trough, from any adhering solution, as a communication of saline matter was sometimes formed between the solution in the receiver and the atmosphere without. When these precautions were attended to, saline solutions over mercury remained as long without crystallizing, as when separated from the atmosphere in the usual mode.

Solutions which completely filled the receivers when placed in the trough, allowed a portion of mercury to enter, by contracting materially as they cooled. A bubble of air could thus be thrown up, without expelling any of the solution from the receiver, and the crystallization determined, without exposing the solution directly to the atmosphere.

The first observation made, was, that solutions of sulphate of soda sometimes did not crystallize at all upon the introduction of a bubble of air, or at least for a considerable time. This irregularity was chiefly observed in solutions formed at temperatures not exceeding