The Collected Mathematical Papers of Arthur Cayley, Volume 10 |
Contents
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Common terms and phrases
9-functions a₁ aa₁ abcde B₁ C₁ coefficients conic consider const constant coordinates corresponding cos² covariants curve de)² deg-order denote determine dH dH dp dq du)² dx dy dy dz equation H exact differential expression factor foregoing formula four given function Hamiltonian system hence homography intersection Journal of Pure letters linear function Messenger of Mathematics nodal line nodes obtain oval parameter partial differential equation plane q dy quadric surface quartic quartic equation quartic functions R₁ radius regard relation respectively roots sin² sn² solution squares substitutions suppose symbols syzygy tangent thence theorem U₁ values variables writing x₁ Y₁ ди
Popular passages
Page 312 - ... externally by the earth's surface, the penumbral curve being then a mere point. It is necessary to consider on the earth's surface a curve which may for shortness be termed the horizon ; viz. this is the curve of contact of the cone, vertex F, circumscribed about the earth ; this is a small circle nearly coincident with the great circle which is the intersection by a plane through the centre of the earth at right angles to the line from this point to the centre of the sun. Regarding F as a point...
Page 605 - A's speaking the truth plt we mean that if A makes the statement, "B says that the event took place," this is to be regarded as a simple statement, and the probability of the truth of the statement is =pi', viz. I think that Martin introduces into his solution a hypothesis contradictory to the assumptions of the question. I remark further that in my solution I assume that the event is of such a nature that, when there is any testimony in regard to it, the probability is determined by that testimony,...
Page 579 - Í4C ^ hence the average required is obviously v^\. . ^ = ттг3507. (Proposed by Professor CAYLEY.) — Show that, for the quadric cones which pass through six given points, the locus of the vertices is a quartic surface having upon it twenty-five right lines ; and, thence or otherwise, that for the quadric cones passing through seven given points the locus of the vertices is a sextic curve.
Page 272 - In the latter paper, I remark that considering the five regular figures drawn in proper relation to each other on the same spherical surface, the only points which have to be considered are 12 points A, 20 points B, 30 points 0, and 60 points Ф.
Page 602 - Shew that the envelope of a variable circle having its centre on a given conic and cutting at right angles a given circle is a bicircular quartic ; which, when the given conic and circle have double contact, becomes a pair of circles : and, by means of the last-mentioned particular case of the theorem, connect together the porisms arising out of the two problems (1) given two conies, to find a polygon of n sides inscribed in the one and circumscribed about the other.
Page 327 - colour-diagram," and the maxim, adopted by Dyck as the motto of his " Gruppentheoretische Studien " : " A group is defined by means of the laws of combinations of its symbols." This ultimate symbolical form of a group is, so to speak, its transcendental essence, which may become incarnate in an endless variety of shapes, such as sets of permutations, geometrical configurations, motions in space, and so on. In the region of pure algebra we may notice the tenth memoir on quantics...
Page 468 - Mémoire sur les fonctions de deux variables et à quatre périodes qui sont les inverses des intégrales ultra-elliptiques de la première classe (Mémoires des savants étrangers, t.
Page 580 - Find at any point of a plane curve the angle between the normal and the line drawn from the point to the centre of the chord parallel and indefinitely near to the tangent at the point...
Page 272 - В the centres of the faces, the points 0 the mid-points of the sides, (viz. each point в is the common mid-point of a side of the dodecahedron and a side of the icosahedron, which sides there intersect at right angles), and the points Ф are points lying by 3's on the faces of the icosahedron, each point Ф of the face being given as the intersection of a perpendicular...
Page 580 - Find the equation of the first negative focal pedal of (1) an ellipsoid, and (2) an ellipse. Solution by PROFESSOR CAYLEY. 1. It is easily seen that if a sphere be drawn, passing through the centre of the given quadric and touching it at any point (x...