strongest of all the ties which bound California so securely to the Union, in the dark days of its struggle for existence. The secession element in Missouri recognized the importance of the line in this respect, and were persistent in their efforts to destroy it. We have seen by what means their purpose was thwarted.

I have always felt that, among the countless evidences of the ordering of Providence by which the war for the preservation of the Union was signalized, not the least striking was the raising up of this remarkable man, to accomplish alone, and in the very nick of time, a work which at once became of such national importance.

This is the man who has crowned his useful career, and shown again his eminently practical character and wise foresight, by the endowment of this College, which cannot fail to be a perennial source of benefit to the country whose interests he has done so much to promote, and which his remarkable sagacity and energy contributed so much to preserve.

We have an excellent rule, followed by all successful designers of machinery, which is, to make provision for the extreme case, for the most severe test to which, under normal conditions, and so far as practicable under abnormal conditions also, the machinery can be subjected. Then, of course, any demands upon it which are less than the extreme demand are not likely to give trouble. I shall apply this principle in addressing you to-day. In what I have to say, I shall speak directly to the youngest and least advanced minds among my auditors. If I am successful in making an exposition of my subject which shall be plain to them, then it is evident that I need not concern myself about being understood by the higher class men and the professors.

The subject to which your attention is now invited is

THE PRINCIPLES AND METHODS OF BALANCING FORCES DEVELOPED IN MOVING BODIES.

This is a subject with which every one who expects to be concerned with machinery, either as designer or constructor, ought to be familiar. The principles which underlie it are very simple, but in order to be of use, these need to be thoroughly understood. If they have once been mastered, made familiar, incorporated into your intellectual being, so as to be readily and naturally applied to every case as it arises, then you occupy a high vantage ground. In this particular, at least, you will not go about your work uncertainly, trying first this method and then that one, or leaving errors to be disclosed when too late to remedy them. On the contrary, you will make, first your calculations and then your plans, with the certainty that the result will be precisely what you intend.

Moreover, when you read discussions on any branch of this subject, you will not receive these into unprepared minds, just as apt to admit error as truth, and possessing no test by which to distinguish the one from the other; but you will be able to form intelligent judgments with respect to them. You will discover at once whether or not the writers are anchored to the sure holding ground of sound principles.

It is to be observed that I do not speak of balancing bodies, but of balancing forces. Forces are the realities with which, as mechanical engineers, you will have directly to deal, all through your lives. The present discussion is limited also to those forces which are developed in moving bodies, or by the motion of bodies. This limitation excludes the force of gravity, which acts on all bodies alike, whether at rest or in motion. It is, indeed, often desirable to neutralize the effect of gravity on machinery. The methods of doing this are, however, obvious, and I shall not further refer to them.

Two very different forces, or manifestations of force, are developed by the motion of bodies. These are

MOMENTUM AND CENTRIFUGAL FORCE.

The first of these forces is exerted by every moving body, whatever the nature of the path in which it is moving, and always in the direction of its motion. The latter force is exerted only by bodies whose path is a circle, or a curve of some form, about a central body or point, to which it is held, and this force is always at right angles with the direction of motion of the body.

Respecting momentum, I wish only to call your attention to a single fact, which will become of importance in the course of our discussion. Experiments on falling bodies, as well as all experience, show that the velocity of every moving body is the product of two factors, which must combine to produce it. Those factors are force and distance. In order to impart motion to the body, force must act through distance. These two factors may be combined in any proportions whatever. The velocity imparted to the body will vary as the square root of their product. Thus, in the case of any given body,

This table might be continued indefinitely. The product of the force into the distance will always vary as the square of the final velocity imparted. To arrest a given velocity, the same force, acting through the same distance, or the same product of force into distance, is required that was required to impart the velocity.

The fundamental truth which I now wish to impress upon your minds is that in order to impart velocity to a body, to develop the energy which is possessed by a body in motion, force must act through distance. Distance is a factor as essential as force. Infinite force could not impart to a body the least velocity, could not develop the least energy, without acting through distance.

This exposition of the nature of momentum is sufficient for my present purpose. I shall have occasion to apply it later on, and to describe the methods of balancing this force, in those cases in which it becomes necessary or desirable to do so. At present I will proceed to consider the second of the forces, or manifestations of force, which are developed in moving bodies--centrifugal force.

This force presents its claims to attention in all bodies which revolve about fixed centers, and sometimes these claims are presented with a good deal of urgency. At the same time, there is probably no subject, about which the ideas of men generally are more vague and confused. This confusion is directly due to the vague manner in which the subject of centrifugal force is treated, even by our best writers. As would then naturally be expected, the definitions of it commonly found in our handbooks are generally indefinite, or misleading, or even absolutely untrue.

Before we can intelligently consider the principles and methods of balancing this force, we must get a correct conception of the nature of the force itself. What, then, is centrifugal force? It is an extremely simple thing; a very ordinary amount of mechanical intelligence is sufficient to enable one to form a correct and clear idea of it. This fact renders it all the more surprising that such inaccurate and confused language should be employed in its definition. Respecting writers, also, who use language with precision, and who are profound masters of this subject, it must be said that, if it had been their purpose to shroud centrifugal force in mystery, they could hardly have accomplished this purpose more effectually than they have done, to minds by whom it was not already well understood.

Let us suppose a body