296 REPORT— 1861. 



(i.e. if it be an expression of the form Ax"), the problem reduces itself to 

 that of the representation of the number A by the form/;. If, again,/, contains 

 as many indeterminates as/, the problem becomes that of the transformation 

 of/ into/. We may add that the problem of improper representation may 

 be made to depend on that of proper representation, by methods analogous 

 to those by which the problem of transformation depends on that of equiva- 

 lence. (See Disq. Arith. art. 284, where Gauss treats of the improper re- 

 presentation of binary by ternarj' quadratic forms.) 



83. It is hardly necessary to state that what has been done towards obtaining 

 a complete solution of these problems is but very little compared with what 

 remains to be done. Our knowledge of the algebra of homogeneous forms (not- 

 withstanding the accessions which it has received in recent times) is far too 

 incomplete to enable us even to attempt a solutioo of them co-extensive with 

 their general expression. And even if our algebra were so far advanced as to 

 supply us with that knowledge of the invariants and other concomitants of 

 homogeneous forms which is an essential preliminary to an investigation of 

 their arithmetical properties, it is probable that this arithmetical investiga- 

 tion itself would present equal difficulties. The science, therefore, has as 

 yet had to confine itself to the study of particular sorts of forms; and of 

 these (excepting linear forms, and forms containing only one indeterminate) 

 the only sort of which our knowledge can be said to have any approach to 

 completeness are the binary quadratic forms, the first in order of simplicity, 

 as they doubtless are in importance. Of all other sorts of forms our know- 

 ledge, to say the least, is fragmentary. 



We shall arrange the researches of which we have now to speak in the 

 following order, according to the subjects to which they refer : — 



1. Binary Quadratic Forms. 



2. Binary Cubic Forms. 



3. Other Binary Forms. 



4. Ternary Quadratic Forms. 



5. Other Quadratic Forms. 



6. Forms of order n decomposable into 7i linear factors. 



The theory of linear forms (i. e. of linear indeterminate equations) we shall 

 refer to hereafter. That of forms containing only one indeterminate will 

 not require any further notice. 



(1) Binary Quadratic Forms. 



84. Instead of confining our attention exclusively to the most recent 

 researches in the Theory of Quadratic Forms, we propose, in the following 

 articles, to give a brief but systematic resume of the theory itself, as it 

 appears in the Disq. Arith., introducing, in their proper places, notices, as 

 full as our limits will admit, of the results obtained by later mathematicians. 

 We adopt this method, partly to render the later researches themselves more 

 easily intelligible, by showing their connexion with the M'hole theory ; but 

 partly also in the hope of facilitating to some persons the study of the Fifth 

 Section of the Disq. Arith., which, probably owing to the obscurity of 

 certain parts of it, is even now too much neglected by mathematicians. 

 This section is composed, as Lejeune Dirichlet has observed (Crelle, vol. xix. 

 p. 325), of two very distinct parts. The results contained in the former of 

 the two (arts. 153-222) are lor the most part those which had been already 

 obtained by Euler, Lagrange, and Legendre ; but they are completed in 

 many respects ; they are derived, in part at least, from different principles, 



