# ELEMENTARY THEOREMS BELATING TO DETERMINANTS

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### Contents

 Section 1 1 Section 2 2 Section 3 3 Section 4 4 Section 5 5
 Section 6 6 Section 7 7 Section 8 29 Section 9 55

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Page vii - The general question therein proposed is, „,To find all the derivatives of any number of functions which have the property of preserving their forms unaltered after any linear transformations of the variables.
Page 49 - Theorem XV. A symmetrical sJcew determinant of an even order does not in general vanish, but the system has foi its inverse a symmetrical skew system.
Page 12 - ... if the whole of a vertical or horizontal row be multiplied by any quantity, the determinant is multiplied by that quantity.
Page 61 - ... equations (so that the variables of each set may be considered as functions of those of the other set) the quotient of the expressions dxdy ... and dudv ... is equal to the quotient of two determinants formed with the functions which equated to zero express the relations between the two sets of variables ; the former with the differential coefficients of these functions with respect to u, v ... , the latter with the differential coefficients with respect to x, y ____ Consequently the notation...
Page 21 - If a determinant of the nth order vanishes^ a system of n homogeneous linear equations, the coefficients of which are the constituents of the given determinant^ may always be established. By Theorem I it also appears that this theorem holds good whether the determinant be resolved according to its vertical or its horizontal rows. Besides the cases noticed in the introductory section, the following 'are examples of this theorem. The condition that three straight lines may be parallel...
Page 29 - And if a, o, w be the three focal chords, 'parallel to the three sides of an inscribed triangle, s another focal chord perpendicular to the major axis, and r the radius of the circle passing through the angular points of the inscribed triangle, there would be found, by a process similar to that used above: In the general formula given above, when the three points are conjugate, that is to say, when the polar of each passes through the other two, we have (2,3) = 0, (3,1) = 0, (1,2) = 0, and the expression...
Page vi - Politechnique tome IX. cahier 16. The next volume of this series contains a paper by Cauchy^ written at the same time, on functions which only change sign when the variables which they contain are transposed. The second part of this paper refers immediately to determinants, and contains a large number of very general theorems. Amongst them is noticed a property of a class of functions closely connected with determinants, first given, so far as I am aware, by fandermonde ; if in the development of...
Page 16 - ... as great as we please, whilst the second factor can be made as nearly equal to An as we please. Hence the product can be made as great as we please. Moreover, if x be positive the sign of the product will be the same as that of An, whilst if a; be negative the sign will be the same as that of An, or the reverse, according as n is even or odd. It follows that in the graphical representation of a rational integral function y=f(x) the curve is everywhere at a finite distance from the axis of x,...
Page 63 - Consider the system of differential equations dx : dy : dz . . = x : Y : z . . (where, for greater clearness, an additional letter z has been introduced). From these we deduce the equivalent system...
Page iv - Vandermonde relating to determinants of the second, third, fourth etc. order. The former, in discussing a system of simultaneous differential equations, has given the law of formation, and shown that when two horizontal or vertical rows (according to the notation of the present work) are interchanged, the sign of the determinant is changed.