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ABCD angles arbitrary axes axis becomes centre chords circle circular combinant common complete condition cones conic conjugate consequently considered constant contain corresponding curve cylinder denoted determined differential direct distance edges effect eliminate envelope equal equation evidently expression faces four functions given gives groups heat Hence homogeneous functions imaginary integral intersect length letters limit locus logarithms means method multiply observed obtained operation opposite origin pair parabola parameters pass places plane points points of contact positive pressure properties quantities reduced reference relation remaining represent respectively result second degree similar solution sphere straight lines substituting suppose surface symbols tangent temperature tetrahedron theorem theory tion touch vanishes variables volume write
Page 141 - On the Theorems in Space analogous to those of Pascal and Brianchon in a Plane, is an admirable specimen of analytical skill.
Page 250 - ... or to part with heat to surrounding matter, until its temperature sinks to T. (3) Let the air be allowed to part with heat so as to keep its temperature constantly T, while it is compressed to such an extent that at the end of the fourth operation the temperature may be S. (4) Let the air be further compressed, and prevented from either gaining or parting with heat, till the piston reaches its primitive position. The amount of mechanical effect gained on the whole of this cycle of operations...
Page 94 - I say there is no such curve line, but I can, in less than half a quarter of an hour, tell whether it may be squared, or what are the simplest figures it may be compared with, be those figures conic sections or others. And then, by a direct and short way, (I dare say the shortest the nature of the thing admits of, for a general one,) I can compare them.
Page 95 - ... according to Boyle's law always, and if the earth were at rest in a space of constant temperature with an atmosphere of the actual density at its surface*.
Page 73 - Hence, in the circle, if d be the angle which gives the circular arc equal to the radius, 2$ is the angle which will give an arc equal to twice the radius, and so on for any number of angles. This is of course self-evident in the case of the circle, but it is instructive to point out the complete analogy which holds in the trigonometries of the circle and of the parabola. Hence the amplitude which gives the difference between the parabolic arc and its subtangent equal to the semiparameter is given...
Page 253 - Vortices, when applied to any fluid whatever, experiencing a cycle of four operations satisfying Carnot's criterion of reversibility (being, in fact, precisely analogous to those described above, and originally invented by Carnot) ; and he thus establishes Carnot's law as a consequence of the equations of the mutual conversion of heat and expansive power, which had been given in the first section of his paper on the Mechanical Action of HeatJ.
Page 275 - ... the foot of the perpendicular let fall from the centre of the wave on the plane of circular contact.
Page 98 - ... xyz, or vice versa ; but in this case the two resultants are not essentially distinct, the one being derivable from the other by mere transference of lines. Cayley then adds, " And in general for any even number of quadratic radicals the two forms are not essentially distinct,! but may be derived from each other by interchanging lines and columns, while for an odd number of quadratic radicals the two forms cannot be so derived from each other, but are essentially distinct.
Page 65 - Tertex may be expressed by the integral f sec 6 dt), 0 being the angle which the normal to the arc at its other extremity makes with the axis, or the angle between the normals drawn to the arc at its extremities. -1- and -r may be called logarithmic plus and minus. As examples of the analogy which exists between the trigonometry of the parabola and that of the circle, the following expressions in parallel columns are given ; premising that the formulae marked by corresponding letters may be derived...
Page 68 - If we call an arc measured from the vertex of a parabola an apsidal arc, to distinguish it from an arc taken anywhere along the parabola, the preceding theorem will enable us to express an arc of a parabola, taken anywhere along the curve, as the sum or difference of an apsidal arc and a right line. Thus, let VCD be a parabola, S its focus, and V its vertex. Let VB=n(w.^), VC=rj(»».x), VD=n(»».a>), and let=A.