designed and constructed. The plate is a photograph of the two forms.

The principle on which the larger one works is identical with what has already been described, that is to say, it consists essentially of a wheel rolling on the paper in such a manner that at any point its plane is inclined to the horizontal, at an angle the tangent of which is always proportional to the corresponding ordinate of the curve which it is desired to integrate. The movement of this rolling wheel actuates a pencil, which draws the required integral curve.

The instrument consists of a large frame NG mounted on wheels HHO in such a manner that it can freely roll in a horizontal direction parallel to itself. On the frame run two jockey carriages, C, D, free to roll vertically on the frame in straight grooves.

The upper jockey D carries the integrating wheel and an adjustable pencil Q, and the lower one an adjustable pointer P, which is for the purpose of tracing out the curve which it is desired to integrate.

The mechanism, which will be understood from a careful inspection of the photograph, is such that, when the pointer is at a distance r above or below the zero position, the integrating wheel is inclined at an angle whose tangent is proportional to x.

Suppose the pointer is at zero, that is to say, is on the base line OX of the curve, which it is desired to integrate. The tiller bar, guided by the wheel B which rolls on the bar E, then compels the plane of the rolling wheel to be parallel to the horizontal base, OX. If the frame is now moved to the right, the rolling wheel rolls on the paper in a direction parallel to the base line, and does not, therefore, move the upper jockey on the frame. But when the pointer is at a distance r above the base line, the tiller bar inclines the integrating wheel at an angle e such that tan ∞ x.

If the frame is now rolled on the paper the integrating wheel rolls in an inclined direction, the tangent of the inclination of which is proportional to x, the distance of the pointer above its zero position. The rolling wheel carries the upper jockey with it, and the vertical movement of the upper jockey, which is read off by means of verniers L, is a measure of the area of the lower curve above the base line to a scale which is determined by the point of attachment of the tangent bar KS.

The pencil, being attached to the upper jockey, draws a curve on the paper which is exactly similar to the curve rolled out by the integrating wheel. The scale of the upper curve is determined by the position in which the clamps R, S, are fixed to the graduated horizontal members N of the frame.

NOTES.

(1) PAGE 13, line 15. To be strictly accurate the 100 feet should be measured along the sloping railway, and not horizontally. When the slope is no greater than the gradient of an ordinary railway the difference between the two measurements is, for practical purposes, inappreciable.

(2) Page 14, line 15. The word "rate" derived from "ratio" does not necessarily embody the idea of time, though this is its common use. Thus we speak of a "rate of wages" implying so much money per hour or per week. On the other hand, it is common to speak of a "piece rate," implying so much money per piece, the latter having no reference to time.

(3) Section 12. There are two meanings commonly attached to the word "zero." Thus when, speaking literally, we say that a man has no money, we mean that he is absolutely without any single coin. On the other hand, we may say that the difference between two sovereigns is nothing, or zero. We do not imply thereby that they are so absolutely equal that no difference in weight could be detected between them, but that for practical purposes each is equivalent in value to twenty shillings. It is probably impossible to imagine two sovereigns so exactly equal in weight that no conceivable method of weighing could reveal a difference between them. The mathematical conception of zero is a sort of compromise between these two ideas. It is not literally "nothing." It would be impossible to conceive or reason about nothing. is not exactly a small quantity so small as to be in pure mathematics there are no such quantities. equations, for instance) and are "neglected" the values are not mathematical equivalents, but only "approximations." They may be more or less close approximations, but they are always approximations. Absolute equations, on the other hand, are such that no conceivable process of measurement or argument could detect any difference whatever between the two sides. Perhaps the best concrete illustration of a mathematical zero is the difference between ++++ etc., to an unlimited number of terms, and unity; or the difference between o'g and I'000. This difference would be strictly described in abstract mathematics as zero." The above discussion may seem rather like hair splitting, but

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On the other hand, it unimportant, because When they occur (in resulting equations or