Sentence Examples

  • A binary form which has a square factor has its discriminant equal to zero.
  • A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n-3 coefficients, together with the determinant of transformation, give us n -2 parameters, and in consequence one relation must exist between any n - I invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its members.
  • The nomenclature of acids follows the same general lines as that for binary compounds.
  • -, reduce s x2ax1 -x10x2 to the form j Oz ON 2 1 1 j 2 i The Binary Quintic.-The complete system consists of 23 forms, of which the simplest are f =a:; the Hessian H = (f, f') 2 = (ab) 2axbz; the quadratic covariant i= (f, f) 4 = (ab) 4axbx; and the nonic co variant T = (f, (f', f") 2) 1 = (f, H) 1 = (aH) azHi = (ab) 2 (ca) axbycy; the remaining 19 are expressible as transvectants of compounds of these four.
  • The process is not applicable with complete success to quintic and higher ordered binary forms. This arises from the circumstance that the simple syzygies between the ground forms are not all independent, but are connected by second syzygies, and these again by third syzygies, and so on; this introduces new difficulties which have not been completely overcome.