near one another between two conjugate Kinetic Foci, proved ultimately equal-If two sides, deviating infinitely little from the third, are together equal to it, they con- stitute an unbroken natural course-Natural course proved not a course of Minimum Action, beyond a Kinetic Focus -A course which includes no Focus conjugate to either extremity includes no pair of conjugate Foci-How many Kinetic Foci in any case-Theorem of Maximum Action -Applications to two Degrees of Freedom-Hamilton's Observation and Experiment-Rules for the conduct of Ex- periments-Residual phenomena-Unexpected agreement or discordance of results of different trials Hypotheses-Deduction of most probable result from a num- ber of observations-Law of Error-Probable Error- Probable Error of a Sum, Difference, or Multiple-Prac- tical application-Method of Least Squares-Methods re- 356-368 369-382 383-398' CHAPTER IV.-MEASURES AND INSTRUMENTS. Necessity of accurate Measurements-Classes of Instruments -Calculating Machines-Angular Measure-Measure of Time-Necessity for a Perennial Standard. A Spring sug- gested-Measure of Length, founded on artificial Metallic Standards-Measures of Length, Surface, Volume, Mass Clock-Electrically controlled Clocks-Chronoscope-Diagonal meter-Cathetometer-Balance-Torsion-balance-Pen- dulum-Bifilar Balance-Bifilar Magnetometer-Absolute 414-437 II. Machine for the Solution of Simultaneous Linear Equations. III. An Integrating Machine having a New Kinematic Principle-Disk-, Globe-, and Cylinder-Integrator. IV. An Instrument for calculating sø (x) ¥ (x) dx, the Integral of the Product of two given Functions. V. Mechanical Integration of Linear Differential Equations of the Second Order with Variable Coefficients. VI. Mechanical Integration of the general Linear Differential Equation of any Order with Variable Coefficients. VII. Harmonic Analyzer-Tidal Harmonic Analyzer-Secondary, ter- DIVISION I. PRELIMINARY. CHAPTER I.-KINEMATICS. 1. THERE are many properties of motion, displacement, and deformation, which may be considered altogether independently of such physical ideas as force, mass, elasticity, temperature, magnetism, electricity. The preliminary consideration of such properties in the abstract is of very great use for Natural Philosophy, and we devote to it, accordingly, the whole of this our first chapter; which will form, as it were, the Geometry of our subject, embracing what can be observed or concluded with regard to actual motions, as long as the cause is not sought. 2. In this category we shall take up first the free motion of point, then the motion of a point attached to an inextensible cord, then the motions and displacements of rigid systems—and finally, the deformations of surfaces and of solid or fluid bodies. Incidentally, we shall be led to introduce a good deal of elementary geometrical matter connected with the curvature of lines and surfaces. point. 3. When a point moves from one position to another it must Motion of a evidently describe a continuous line, which may be curved or straight, or even made up of portions of curved and straight lines meeting each other at any angles. If the motion be that of a material particle, however, there cannot generally be any such abrupt changes of direction, since (as we shall afterwards see) this would imply the action of an infinite force, except in the case in which the velocity becomes zero at the angle. It is useful to consider at the outset various theorems connected VOL. I. 1 point. Motion of a with the geometrical notion of the path described by a moving point, and these we shall now take up, deferring the consideration of Velocity to a future section, as being more closely connected with physical ideas. Curvature of a plane curve. 4. The direction of motion of a moving point is at each instant the tangent drawn to its path, if the path be a curve, or the path itself if a straight line. 5. If the path be not straight the direction of motion changes from point to point, and the rate of this change, per unit of length of the curve (according to the notation below below) is called the curvature. To exemplify this, suppose two tangents drawn to a circle, and radii to the points of contact. The angle between the tangents is the change of direction required, and the rate of change is to be measured by the relation between this angle and the length of the circular arc. Let I be the angle, c the arc, and p the radius. We see at once that (as the angle between the radii is equal to the angle between the tangents) pl = c, I 1 and therefore Hence the curvature of a circle is inс P versely as its radius, and, measured in terms of the proper unit of curvature, is simply the reciprocal of the radius. 6. Any small portion of a curve may be approximately taken as a circular arc, the approximation being closer and closer to the truth, as the assumed arc is smaller. The curvature is then the reciprocal of the radius of this circle. If 80 be the angle between two tangents at points of a curve distant by an arc ds, the definition of curvature gives us at once as its measure, the limit of when ds is diminished without 80 88 limit; or, according to the notation of the differential calculus, But we have do ds' if, the curve being a plane curve, we refer it to two rectangular of a plane axes OX, OY, according to the Cartesian method, and if 0 denote Curvature the inclination of its tangent, at any point x, y, to OX. Hence curve. 0 = tan- dy ; and, by differentiation with reference to any independent variable t, we have Although it is generally convenient, in kinematical and kinetic formulæ, to regard time as the independent variable, and all the changing geometrical elements as functions of it, there are cases in which it is useful to regard the length of the arc or path described by a point as the independent variable. On this supposition we have 0 = d (ds3) = d (dx2 + dy3) = 2 (dx d ̧3x + dy d ̧3y), where we denote by the suffix to the letter d, the independent and using these, with ds2 = dx2 + dy3, to eliminate dx and dy 1 _ {(d,3y)' + (d,2x)°}*. = ; or, according to the usual short, although not quite complete, curve. 7. If all points of the curve lie in one plane, it is called a Tortuous plane curve, and in the same way we speak of a plane polygon or broken line. If various points of the line do not lie in one plane, we have in one case what is called a curve of double |