292 REPORT — 1861. 



Report on the Theory of Numbers. — Part III. ByU. J. Stephem 

 Smith, M.A., F.R.S., Savilian Professor of Geometry in the Univer- 

 sity of Oxford. 



(B) Theory of Homogeneous Forms. 



79. Problem of the Representation of Numbers. — A rational and integral 

 homogeneous function (a quantic according to tlie nomenclature introduced 

 by Mr. Cayley), of wliich the coefficients are integral numbers, is, in the 

 Theory of Numbers, termed a form (Disq. Arith. art. 266). The form 

 is linear, quadratic, cubic, biquadratic or quartic, quintic, &c., accord- 

 ing to its order in respect of the indeterminates it contains ; and binary, 

 ternary, quaternary, &c., according to the nnmber of its indeterminates. 

 Thus a;- -h?/- is a binary quadratic form, x^ + ^f + z^ — Sxyz a ternary cubic 

 form. A form is considered to be given, when its coefficients are given 

 numbers; and a number is said to be represented by a given form, when 

 integral values are assigned to the indeterminates of the form, such that the 

 form acquires the value of the number. If the values of the indeterminates 

 are relatively prime, the representation is said to be primitive ; if they 

 admit any common divisor beside unity, it is a denved representation. 

 Thus 13 and 8 can be represented by x'+f; for 3' + 2^=13, 2- + 2'-=8; 

 and the first of these representations is primitive, the second is derived. 

 The fir^t general problem, then, that presents itself in this part of the 

 Theory of Numbers, is the following, "To find whether a given number 

 is or is not capable of representation by a given form, and, if it is, to find 

 all its representations by that form." The number of different representations 

 of a given number by a given form may be either finite or infinite ; in the 

 former case the complete solution of the problem of representation consists 

 in the actual exhibition of the different sets of values that can be given to 

 the indeterminates of the form: in the latter case it consists in assigning 

 general formula;, in which all those values are comprised. It is in either 

 case sufficient to consider primitive representations only ; for if the given 

 form/ be of order »n, and the given number N be divisible by the m'* powers 

 d/", rf„'", , the derived representations of N by / coincide Avith the 



N N 

 primitive representations of -j^, -j—, by the same form. 



80. Problems of the Transformation and Equivalence of Forms. — A form 

 f.f^x\, x .^, .... .r'„) is said to be contained in another form ffx^^ x.^-, .... Xr^, 

 when/j arises from/ by a linear transformation of the type 



^i=«i,r< + «i,2^'2 + +«i,y„' 



''^„ = ««,l^'l + ««,2''«^'2 + +««,«<' 



in which the coefficients a.^■ are integral numbers and the determinant is 

 different from zero*. This transformation we may, for brevity, describe as 

 the transformation \a\. When \a\ is a unit-transformation, i.e. when the 

 determinant of |a | is a positive or negative unit, the inverse transformation 



* Gauss says that/2 is contained inyi, even when the determinant of transformjition is 

 zero (Disq. Arith. art. 215). But we slrall find it more convenient to retain the restriction 

 specified iu the text. 



