produced entirely by the attraction which one body (the sun or the moon) exerts separately upon every separate particle of the earth. Upon these I have already spoken, (see page 175); and there will be no need for me to detain you further at present, because you will have been sufficiently aware that there is general conformity between the results which I obtained upon Newton's theory and the results obtained by actual observation. With regard to the numerical agreement, I shall make some remarks presently, when I speak of the mass of the moon.

I will now speak of the ellipticity of the earth; and this, it will be found, is a case in which it is necessary to consider every particle as attracting every particle. First of all you will remember that when the hoop, Figure 23, is put in motion round the vertical spindle, it changes its form. Now in order to explain this, there is a term commonly used which I believe I have not in these lectures hitherto uttered; the reason is, that I do not like it; I allude to the term " centrifugal force." In order to explain why this hoop expands horizontally when it is whirled round the vertical axis, I must recall to your minds the first law of motion. The first law of motion as applied to the hoop is this: if the part a of the hoop is put in motion horizontally, it would go on in a horizontal straight line if it could. No matter what may be the nature of the force which puts a in motion, it has no tendency to move in a circle; and if it were set free, as a stone from a sling, it would immediately fly off in a straight line. And by motion in a straight line, it would go further and further still from the central bar. In order to keep it at the same distance from the central bar, a restrain

ing force is necessary. The term "centrifugal

R

force" has been used to express the tendencies of the various parts of that hoop to acquire greater distances from the central bar. It is a bad term, because in reality there is no force. Perhaps it would be better to say "centrifugal tendency," tendency to recede from the centre, which will in all cases require a force to control it. Now this centrifugal tendency tends to change the figure of the earth; but the consideration of the centrifugal tendency alone is not sufficient to give us the means of calculating what the form of the earth will be.

Newton was the first person who made a calculation of the figure of the earth, on the theory of gravitation. He took the following supposition as the only one on which his theory could be applied : he assumed the earth to be a fluid, or at least to be so far fluid in all parts below the surface, that its form would be the same as if it were entirely fluid. This fluid matter he assumed to be equally dense in every part, so that it was composed of no heavier matter at the centre than at the circumference. For trial of his theory, he supposed the fluid earth to be a spheroid; he then computed the attraction of the whole spheroid upon every one of its component particles of fluid; with this he combined the centrifugal tendency; and then he examined whether, by giving a proper degree of ellipticity to the assumed spheroid, the forces computed on this supposition would be such as would keep the fluid in the spheroidal form which he had supposed to be the earth's form. Now upon the theory of gravitation it is evident that the attraction of a sphere is not the same thing as the attraction of a spheroid. It is necessary to compute what the attraction of this spheroid is, before we can enter into the effect of its

combination with the centrifugal tendency. This is the result: suppose that the spheroid AB, Figure 61, is not revolving at all; still even in that case the attraction of the spheroid upon a body at the part A of the earth is greater than the attraction upon a body at the part B of the earth. But besides this, when we suppose the earth to revolve round the axis Aa, there is the centrifugal tendency of which I have spoken, which does not affect the body at the part A of the earth in the axis of rotation, but which affects the body at B at a great distance from the axis of rotation. We have to consider then that at the Poles of the earth there is an attraction which may

be computed when we assume that the earth is in the form of a spheroid; and at the equator there is an attraction which may also be computed, and is found to be smaller than that at the Pole, and which is still further diminished by the centrifugal tendency.

Thus the whole effective attraction at the Pole is sensibly greater than the whole effective attraction at the equator. This is not unfrequently expressed

by saying that "a body weighs more at the Pole than at the equator." And this statement is correct, if it be received with the proper caution. If we carried a pair of scales with proper weights from the Pole to the equator, the same weights which balanced a stone at the Pole would balance it at the equator, because the effect of gravity on both is altered in the same degree. But if we carried a spring-balance from the Pole to the equator, the spring would be more bent by the weight of the same stone at the Pole than at the equator. There is also another effect, to which I shall shortly allude, that a stone would fall further in one second at the Pole than at the equator.

Having computed the effective attractions at the Pole and at the equator, we must now examine what is the consideration to be applied in order to discover whether, with a certain supposition of ellipticity of the earth, this homogeneous fluid will be in equilibrium. The way in which Sir Isaac Newton proceeded is the same as that adopted by every other person who treats of the theory of fluids. You may conceive a cylindrical tube AE, open at both ends, to be put down from the Pole to the centre. Suppose you put down a similar pipe BE from the equator to the centre; and suppose that they communicate at the centre E-these imaginary pipes will not at all disturb the state of rest of the fluid, if it be at rest-by means of each of these pipes we shall ascertain the state of pressure of the fluid at the centre E. By the "state of pressure" I mean the measure of that compression of the fluid at E, which would enable it to burst any shell that enclosed it at E, if there were no opposing pressure on the outside of the shell; and this measure is to

be understood as expressed by so many pounds per square inch; just as we measure the pressure of water in the cylinder of a Bramah's press, by so many pounds per square inch, meaning by that, the pressure on every square inch of its case, tending to burst it.

Now in order to find the pressure produced by the fluid in the column AE, it is not sufficient to know the length of that column, but we must also know the attraction which acts on every part of it. In ordinary cases we speak of the pressure of "a head of water," and we measure it by the depth of the water, and that measure is accurate, because gravity acts equally on all the water in such depths as we have to treat of in ordinary cases; but if there were any part of the water on which gravity did not act at all, that part would add nothing to the pressure; or if there were any part on which gravity acted with only half its usual force, that part would contribute only half its proportion to the pressure. We must therefore ascertain, not only the lengths of different parts of the columns of fluid AE and BE, but also the proportions of the attractions acting on those different parts.

Now we have just seen that the attraction, as diminished by the centrifugal tendency, is less at B than at A; and I may now state as a result of mathematical investigation, that the attraction diminished by the centrifugal tendency is less at the middle of EB than at the middle of EA, and so at every corresponding part of their lengths. Therefore when we estimate the pressure produced by the fluid in the column AE, we have to consider that there is a short column of fluid, of which every part is pulled downwards by a large attraction; and for the column