Quadratic Forms: and Their Classification by Means of Invariant-Factors, Paperback/J. G. Leathem

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Description IN writing this tract, the chief difficulty has been to compress the material so as not to entirely outrun the prescribed limits of space. 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 Darboux had been adopted, a long and rather tedious discussion would have been needed for certain determinantal theorems (Arts. 24, 25), 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 account 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 determinant are known (see Arts. 2, 16, 19, 22). I hope that a frequent appeal to geometry may serve to make the algebra more easily understood. The omission of any account of Weierstrass\'s and Darboux\'s methods would be a serious blot, if the tract were intended to be exhaustive rather than suggestive; in particular Darboux\'s treatment of the case of unequal rootsl must always be regarded as a model of algebraical elegance. But accounts of these methods are already available to a certain extent (for references, see Art. 38); and consequently an exposition is less necessary here. I have devoted Chapter V to an exhibition of some applications of the theory; these may serve to convince the reader of its utility; and a glance at the table given in Art. 23 will shew that families containing more than four variables could not be exhaustively classified without the aid of invariant-factors. Indeed, even the case of four variables was not