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already applied becomes binary quantic calculate called canonical form coefficients columns combined common condition connected considered constants constituents contain contravariant corresponding covariant cubic curve degree denote derived determinant differential discriminant divided divisible eliminant equal equations evidently example explained expressed factor follows formula four fourth give given Hence Hessian homogeneous homogeneous function identical imaginary intersection invariant Jacobian last article latter Lesson linear manner method multiplied negative obtained operate original plane points positive Prof proved quadratic quantic quantities reduced relation remaining respectively result roots rows satisfy seen sets side square factor substitute suffixes suppose surface symbol symmetric functions ternary theorem theory third tions transformed values vanish variables weight whence write written
Page 288 - If two rows (or columns) are interchanged, the sign of the determinant is changed.
Page 239 - ... some given screw a, we then obtain a fifth homogeneous equation containing the co-ordinates of 0 in the third degree. The determination of the ratios of the six co-ordinates ft, ... ft is thus effected by five equations of the several degrees I, m, n, r, 3. For each ratio we obtain a system of values equal in number to the product of the degrees of the equations, ie to Slmnr. This is accordingly a major limit to the number of points in which in general a pierces the surface, that is to say, it...
Page 47 - P3 is the product of the squares of the differences of the roots. Thus the product of the squares of the differences of all the roots of an equation can be exhibited as a determinant, the constituents of which are known in terms of the coefficients of the given equation, for s r can be expressed in terms of the coefficients. 398. Suppose we have to find the values of the n unknown quantities...
Page 219 - Article ; for the product of the squares ot the differences of all the roots is made up of the product of the squares of the differences of the roots of...
Page 48 - X = 0 has as many pairs of imaginary roots as there are variations in the series of signs of' the leading terms of the functions these being supposed to diminish in degree regularly by unity.
Page 104 - UK' >lt.. a function of the coefficients of a quantic such that, if the quantic is linearly transformed, the same function of the new coefficients is equal to the first function multiplied by some power of the modulus of transformation.
Page 29 - A minor of the order m formed out of the inverse constituents is equal to the complementary of the corresponding minor of the original determinant A multiplied by the (m — 1)'* power q/"A.
Page 55 - The terms on the right-hand side which are equidistant from the beginning and the end are equal ; therefore by rearranging and dividing by 2 we obtain Now S t , &„...
Page 19 - On account of the importance of this theorem, we give another proof, founded on our first definition of a determinant.
Page 95 - The discriminant of the product of two quantics is equal to the product of their discriminants multiplied by the square of their eliminant. For the product of the squares of differences of all the roots evidently consists of the product of the squares of differences of two...