with the aid of known quantities, which was the object to be gained.

This forms but the first part of our theory, and there remains for us to see how to determine the initial depression H. This determination will be the object of the next paragraph; but before undertaking it, I have pleasure in showing the wonderful parallelism which exists between the preceding theory and that which I have proposed in the two former parts of this study for the volumogen ventilator. I have said already that to pass from the one to the other it is sufficient to change the word "volume" to that of "depression," and conversely. This will be clearly seen from the following.

Volumogen.

In theory the volumogen ventilator should give a constant volume of air, equal to that which it affords when working freely on the atmosphere. In reality the volume obtained is always less than the theoretic volume, because of the unavoidable play of the joints of the machine, which gives place to a reentry direct from the exterior air.

If W represents the theoretical volume, V. the reentry of air, and V the effective volume, I may write the equation

Deprimogen.

In theory the deprimogen ventilator should afford a constant depression equal to that which it reaches when acting upon a closed space. In reality the depression obtained is always less than this initial depression. A great part of this latter will be found to be absorbed by the frictions and losses of active power in the air traversing the machine.

If H represents the initial depression, h, the part absorbed by the frictions, h the effective depression, I may write the equation—

h = H - h。;

a being the equivalent orifice of the mine, we will call o the equivalent of the passage,

We see the parallelism is perfect, and also that their characteristics are absolutely opposite. Thanks to this remarkable reciprocity, the two theories render each other more clear; and I may say that more than once the volumogen ventilator has assisted me to understand the deprimogen. In reality, it is not between the volume and the depression that the reciprocity is established, but between the volume and the square root of the depression. But that changes nothing in the curious result which I am about to show.

Of the two formulæ which we are about to establish, as well for the volumogen ventilator as for the deprimogen, the most important is evidently that which gives the volume of air yielded per second. This volume is most especially interesting. The curve expressed by this formula, the equivalent orifice being drawn as abscissæ and the volume of air as ordinates, characterises most distinctly the ventilator to which it is applied, and furnishes a very simple and sure means to compare different machines. It is this curve which the Commission of Gard has set itself to determine for the six ventilators submitted to their tests. They have given it the name of the characteristic curve of the ventilators.

2. DETERMINATION OF THE INITIAL DEPRESSION FOR VENTILATORS WITH CENTRIFUGAL FORCE.

I preserve the name of Initial Depression for that value of the depression which is shown when the mine is shut off and rests virtually constant when we do away with the friction of the air. We know that in the volumogen ventilator the mechanical combinations which give place to the volume

engendered, differ absolutely in one machine from those in another, and require each time, for the determination of the volume, special methods of calculation. It is the same for deprimogen ventilators; each type of machine produces the depression by particular methods, each requiring their special calculation. But I will not occupy myself here except with the centrifugal ventilator, the most perfect and therefore the most interesting of all. I will study first the uncovered ventilators with centrifugal force, such as were made by the first inventors (Combes, Letoret, Lambert, &c.), to whom the cover appeared, no doubt, the greatest obstacle to the free escape of the air into the atmosphere. This manner of looking at it, very plausible at first sight, is in reality quite erroneous. We know that to M. Guibal belongs the honour of having first shown that the cover is indispensable for making the ventilators develop their full useful effect. To-day almost every ventilator at work on the collieries possesses this ingenious addition. In order to proceed with regularity we must begin with those which have not got them.

To avoid any difficulty in the establishment of my calculations, I will suppose that the air of the mine before reaching the ouïe is expanded in a large chamber where its speed may be considered almost nothing. In reality, this vestibule will be more useless than useful, but for our theory it has the advantage of placing the ventilator between two motionless atmospheres, the air being drawn from one and thrown into the other. The situation is clear, and the analysis equally so.

I propose, then, in the course of this discussion, to consider only an ideal ventilator, presenting no play, nor shocks, nor losses of quantity of any kind. In practice, of course, it is very far from attaining this ideal, the necessity of a simple construction, so as to work with certainty, obliging us to preserve numerous imperfections in detail; but to introduce these imperfections in the calculation would be to

expose ourselves to inextricable complications. It is much better to go straight to the theoretical result.

The ventilator with centrifugal force without cover is represented in its essential features by Fig. 2, the vanes as drawn showing the profile somewhat bent and approaching the ouïe in a manner to afford the desired inclination, so that the air penetrates without shock between these spaces.

These vanes are sufficiently numerous to assure the regular drawing of the air from the centre to the periphery without eddies or disturbances. I will call r the radius of the ouïe or inlet, R that of the external circumference, w the angular speed of rotation. The result, w R, expresses the absolute speed of the extremities of the vanes. This speed plays a preponderating rôle in all this discussion. I will call it the tangential speed, and I will mark it by the letter u.

The depression produced by such a machine will result from a series of actions, some additive and some subtractive. We may, then, determine them successively and in their order, so as to draw the balance.

First, the motionless air in the vestibule is brought