# Cayley sentence example

cayley

- Two methods of treatment have been carried on in parallel lines, the unsymbolic and the symbolic; both of these originated with Cayley, but he with Sylvester and the English school have in the main confined themselves to the former, whilst Aronhold, Clebsch, Gordan, and the continental schools have principally restricted themselves to the latter.
- A theory of matrices has been constructed by Cayley in connexion particularly with the theory of linear transformation.
- This theorem is due to Cayley, and reference may be made to Salmon's Higher Algebra, 4th ed.
- As modified by Cayley it takes a very simple form.
- Cayley, however, has shown that, whatever be the degrees of the three equations, it is possible to represent the resultant as the quotient of two determinants (Salmon, l.c. p. 89).Advertisement
- Such quantics have been termed by Cayley multipartite.
- Observe the notation, which is that introduced by Cayley into the theory of matrices which he himself created.
- It is on a consideration of these factors of t that Cayley bases his solution of the quartic equation.
- The ternary cubic has been investigated by Cayley, Aronhold, Hermite, Brioschi and Gordan.
- Sylvester, Cayley and MacMahon succeeded, by a laborious process, in establishing the generators for 0=5, and 0=6, viz.: 5 15 531 1 -z 2.1-z 3.1-z 4.1-z 5 ' 1-z2.1-z3.1-z4.1-z5.1-z6' but the true method of procedure is that of Stroh which we are about to explain.Advertisement
- The germs of the theory of determinants are to be found in the works of Leibnitz; Etienne Bezout utilized them in 1764 for expressing the result obtained by the process of elimination known by his name, and since restated by Arthur Cayley.
- And one of Hamilton's earliest advances in the study of his system (an advance independently made, only a few months later, by Arthur Cayley) was the interpretation of the singular operator q()q1, where q is a quaternion.
- The method is essentially the same as that developed, under the name of " matrices," by Cayley in 1858; but it has the peculiar advantage of the simplicity which is the natural consequence of entire freedom from conventional reference lines.
- The formation of the tables of a planet has been described by Cayley as " the culminating achievement of astronomy," but the gigantic task which Newcomb laid out for himself, and which he carried on for more than twenty years, was the building up, on an absolutely homogeneous basis, of the theory and tables of the whole planetary system.
- Cayley's screws were peculiar, inasmuch as they were superimposed and rotated in opposite directions.Advertisement
- Degen in 1816 and Ottoris Sarti in 1823, followed Cayley at moderate intervals, constructing flying models on the vertical screw principle.
- All the models referred to (Cayley's excepted') were provided with rigid screws.
- For the caustic by refraction of parallel rays at a circle reference should be made to the memoirs by Arthur Cayley.
- The extension to curves of any given deficiency D was made in the memoir of Cayley, " On the correspondence of two points on a curve, " - Pore.
- The farreaching discoveries of Sylvester and Cayley rank as one of the most important developments of pure mathematics.Advertisement
- Cayley gave the formula E + 2D = eV + e'F, where e, E, V, F are the same as before, D is the same as Poinsot's k with the distinction that the area of a stellated face is reckoned as the sum of the triangles having their vertices at the centre of the face and standing on the sides, and e' is the ratio: " the angles subtended at the centre of a face by its sides /2rr."
- 1112 which Cayley denotes by (a, b, c, ...)(xi, x2)n (i),(2)Ã¯¿½Ã¯¿½Ã¯¿½ being a notation for the successive binomial coefficients n, 2n (n-I),....
- An important fact, discovered by Cayley, is that these coefficients, and also the complete covariants, satisfy certain partial differential equations which suffice to determine them, and to ascertain many of their properties.
- In England, multiple algebra was developed by j ames Joseph Sylvester, who, in company with Arthur Cayley, expanded the theory of matrices, the germs of which are to be found in the writings of Hamilton (see above, under (B); and Quaternions).
- The scope of his researches was described by Arthur Cayley, his friend and fellow worker, in the following words: "They relate chiefly to finite analysis, and cover by their subjects a large part of it - algebra, determinants, elimination, the theory of equations, partitions, tactic, the theory of forms, matrices, reciprocants, the Hamiltonian numbers, &c.; analytical and pure geometry occupy a less prominent position; and mechanics, optics and astronomy are not absent."Advertisement
- Cayley gave a matrix algebra defining addition, multiplication, scalar multiplication and inverses.