Quadratic Forms and Their Classification by Means of Invariant Factors #232188

The theory is developed in an order which may seem unusual to readers already acquainted with other methods of treatment; but my object has been to obtain a fairly complete account in the minimum of space. If the methods of Weierstrass or Barboux had been adopted, a long and rather tedious discussion would have been needed for certain determinantal theorems (arts. 24, before the real problem of reduction could have been attacked. Further, the singular case would then have required an entirely separate discussion, of which the only satisfactory account1 is both involved and laborious. Both of these objections are avoided by the method used here, which is due in substance to Kronecker. And, in addition, the method lends itself to geometrical explanations (see Arts. 1, 13, 17 and Appendix) and is well adapted for the actual reduction of numerical examples, when once the roots of the fundamental deter minant are known (see Arts. 2, 16, 19, I hepe that a frequent appeal to geometry may serve to make the algebra more easily understood.