J. J. SYLVESTER, M.A., F.R.S., PROFESSOR OF MATHEMATICS IN THE ROYAL MILITARY ACADEMY, N. M. FERRERS, M.A., FELLOW OF GONVILLE AND CAIUS COLLEGE, CAMBRIDGE: ASSISTED BY G. G. STOKES, M.A., F.R.S., LUCASIAN PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE; A. CAYLEY, M.A., F.R.S., LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE; AND M. HERMITE, CORRESPONDING EDITOR IN PARIS. ὅ τι οὐσία πρὸς γένεσιν, ἐπιστημὴ πρὸς πίστιν καὶ διάνοια πρὸς εἰκασίαν ἔστι, LONDON: JOHN W. PARKER, SON, & BOURN, WEST STRAND. 1861. W. METCALFE, PRICE FIVE SHILLINGS. GREEN STREET, CAMBRIDGE. Notes on the Higher Algebra. By James Cockle. (Concluded from On some Applications of Algebra to the Theory of Covariants. On the Criteria of Maxima and Minima of Functions of Two Variables. On some General Theorems in the Calculus of Operations and their On Certain Properties of Prime Numbers. By Rev. J. Wolstenholme. On Coaxial Circles. By John Casey, Trinity College, Dublin On the Variations of the Node and Inclination in the Planetary Notes on Lambert's Angles. By William Walton, M.A., Trinity College NOTICES TO CORRESPONDENTS. The following Papers have been received: Mr. WILLIAM WALTON, "On Certain Analytical Relations between Conjugate Wave-Velocities, Ray-Velocities, and Planes of Polarization;" "On Östrogradsky's Hydrostatical Shell;" "Note on Des Cartes' Rule of Signs ;" and "On the Discontinuity of the Intrinsic Equations to Curves." Mr. GEORGE PAXTON YOUNG, "Determination of the Forms of the Roots of Solvible Quintic Equations whose Coefficients are Functions of a Variable." Mr. JAMES W. WARREN, "On the Internal Pressures within an Elastic Solid." Mr. A. S. HERSCHEL, "Sir Wm. Hamilton's Icosian Game;" and "A Geometrical Curiosity." Mr. JAMES COCKLE, "Abstract of Sir W. Rowan Hamilton's Exposition of Abel's Argument.' Mr. J. BLISSARD, "On the Generalization of Certain Trigonometrical Formulæ." Mr. JOHN CASEY, "M'Cullagh's Property of a Self-Conjugate Triangle and Sir W. Hamilton's Law of Force for a Body describing a Conic Section, Demonstrated Geometrically. P. J. H., "Determination of the Foci of the Conic Sections." Mr. MICHAEL ROBERTS, "On the Covariants of a Binary Quantic of the nth Degree. Mr. N. M. FERRERS, "Notes on Tetrahedral and Quadriplanar Coordinates ;" and "On Certain Properties of the Tetrahedron.' Mr. WILLIAM SPOTTISWOODE, "On Petzval's Asymptotic Method of Solving Differential Equations.' Notice to Subscribers.-The first four volumes of the 'Journal,' in bds., or any of the back numbers, can be had on application at the Printing-Office, Green Street, Cambridge. THE QUARTERLY JOURNAL OF PURE AND APPLIED MATHEMATICS. I NOTES ON THE HIGHER ALGEBRA. (Concluded from Vol. IV. p. 57.) NOW propose to consider certain points in the theory of cyclical functions, of Abelian and other cubics, of the second Eulerian quintic, and of the higher prime equations; and to give an exposition of M. Hermite's argument respecting equations of the fifth degree, with a few observations on that of Sir W. R. Hamilton. ON CYCLICAL FUNCTIONS. The theorem which Mr. Harley has been pleased to notice in Art. 3 of his paper "On the Theory of Quintics,' (Quarterly Journal, Jan., 1860) is but one of a class of similar propositions. Let n-1 n (w, x') = x," + wx," + w2x," +...+w"1x", where is an unreal nth root of unity, n being a prime number, and x1, X 29 are the roots of n fx = 0, an algebraic equation of the nth degree put under the usual form. Then (w, x').(w"-1, x') = Σ' {x".(w"-1, x')} VOL. V. = Σ' {x".(w, x')}, B where is the cyclical symbol of Mr. Harley. Hence, adopting the latter form, we have (w, x2). (w1 ́1, xo) + (w”−1, x′). (∞, ∞3) + (w3, x').(w”-2, x°) + (w”-2, x').(w2, xo) + n+1 n+1 ...+ (w +, x"). (wTM*, x') + (w, x). (w, x′′) = Σ' [x°. {(w, x') + (w2-1, x') + (w3, x′) + (w′′-2, x′) = = n-1 +1 +...+ (w, x") + (wTM*, ∞")}] · Σ' [xa {(w, x') + (w2, x3) + (w3, x') + ...+ (w” ̄1, x')}] = Σ' [x,' {(n − 1) x,' — x - x' -... - x„"}] = (n − 1) Σ'x ̧*** — Σ'x ̧° (x,”+x ̧”+...+x„”) = (1 - 1) Σχ – Σκα Making s=r, we have 2-1 #+1 (w, x*).(w” ̄1, x*) + (w3, x').(w”-, x') +...+ (w, x2). (w 2, x3) From the structure of these functions we perceive that when s=r1 they are critical, and leading coefficients (or sources) of covariants. They will probably all be found to have a place in the theory of the higher equations. Indeed, in the case of s=r= 1 the theory of Euler, which for quintics (see Art. 1 of Mr. Harley's paper) gives (w, x).(w1, x) + (w3, x).(w3, x) = 25P, indicates as much. These critical functions, I may add, seem to have their application in the theory of elimination. And if we transform fr=0 to an equation in y, in which y" + Ay"-1 + By"-2 + Cy"-3+...+ Uy+V=0, the result of the elimination of P between A=0 and B=0 is obtained by simply writing down the critical quadratic function, previously expunging from A and B the terms into which P enters: thus In like manner the result of the elimination of P between A=0 and C=0 is given by 3 3n2 C-3n (n − 2) Br_Ap_+ (n − 1) (n − 2) Ap ̧3 = 0. The results of the elimination of P between A=0 and D=0, B=0 and C=0, &c. require special consideration, inasmuch as the number of distinct critical functions of the fourth and higher degrees is not, as in the case of those of the second and third degrees, one only. ON ABELIAN AND OTHER CUBICS. Mr. Jerrard (Phil. Mag., S. 4, Vol. III., pp. 459-460) citing Legendre's Théorie des Nombres, 3rd edition, Vol. II., p. 438, adverts to an antagonism between the results, at which Legendre had arrived, and those of Abel. That Mr. Jerrard is right in supposing that Legendre has overlooked the existence of an equation of condition will appear from the following theorem. The point left doubtful by Mr. Jerrard (b. p. 459, note) is here decided: b, vanishes, that is to say, the rational function Ox does not contain x2. In order that a cubic may be an Abelian, it is necessary, and sufficient, that its quadratic critical function should vanish. or Let fx = 0, x3 + ax2 + bx + c = 0, be the cubic, and x, and x, two of its roots. Then Now in order that the expression under the radical may be rational, we must have |