# Oxford, Cambridge and Dublin Messenger of Mathematics

Macmillan and Company, 1886 - Mathematics
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Page 43 - A cos 6 = cos a cos c + sin a sin c cos B cos c = cos a cos 6 + sin a sin 6 cos C Law of Cosines for Angles cos A = — cos B...
Page 11 - Any function of the coefficients of a quantic is called an invariant, if, when the quantic is linearly transformed, the same function of the new coefficients is equal to the old function multiplied by some power of the modulus of transformation; that is to say, when we have \$ (a', 6', c', &c.) = Ap<£ (a, J, c, &c.).
Page 29 - Interpretation of the equation y = — —,x -f- 2j/'. If this represents the tangent to an hyperbola referred to rectangular asymptotes, — —, is the tangent of the angle which the tangent to the curve makes with the axis of x ; and 2j...
Page 180 - If fig. 179 were spun about OA, what figure would be generated (i) by the circle, (ii) by AP, (iii) by PQ? Hence find the locus of the points of contact of tangents from a fixed point to a fixed sphere. Ex.
Page 78 - ... 37. NOTE ON SCHWARZIAN DERIVATIVES. [Messenger of Mathematics, XV. (1886), pp. 74 — 76.] READING with great pleasure and profit Mr Forsyth's masterly treatise on Differential Equations (in my opinion the best written mathematical book extant in the English language), it occurred to me to find an easy proof of the fundamental and striking identity concerning Schwarzian derivatives, from which all others are immediate consequences, namely (y, x) — (z,x) = IT-) (y, 2), where one of which is,...
Page 35 - WILTON, Trinity College. [Received 9 November 1928, read 20 May 1929.] 1. Let...
Page 79 - ... same thing if y and z functions of x when expressed as functions of x (any function of x) are written y, z', then (y, x') — (z>, x') is identical with (y, x)— (z, x), save as to a factor which depends only on the form of the substitution of x for x. Hence to all intents and purposes, any function of the differences of the Schwarzian derivatives of any system of functions of the same variable, in respect thereto, is (in a sense comprising, but infinitely transcending the sense in which that...
Page 80 - ... highest index of differentiation which such reciprocant contains its order, and the number of factors in each term its degree. Then in any reciprocant so formed the degree is always just one unit less than the order : but as a matter of fact the function so obtained is in general not irreducible, so that its degree may be depressed, and it becomes a question of much interest to form the scale of degrees of reciprocants of this sort. For the orders 2, 3, 4, 5, 6 the degrees in question are respectively...
Page 92 - ... or even according as the smallest number of letters other than a in any of its terms is odd or even. Thus the character of a reciprocant whose leading term is a*e, or ab'e, or abce is odd ; that of one whose leading term is abe or abf is even, as is also that of the remarkable reciprocant bd — oc2 in which no power of a appears.
Page 30 - The asymptotes, if all real, meet the curve again in three finite points, which lie in a right line F, and the product of the distances of any point of the curve from the three asymptotes is in a constant ratio to its distance from the line F. In equation (2) let BDF coincide, and the equation becomes (4) ACE - B2 = 0, an equation also containing nine constants. A, C, E are (Art. 42) tangents at points of inflexion, which lie on a right line B, and we have the theorem proved...