S3 MATHEMATICS, RECENT TERMINOLOGY IV. 



MATHEMATICS, RECENT TERMINOLOGY IN. 



As an instance of the employment of the notation of inatricrs for 

 another purpose, take 



r a, b, e ) (*, y, s) ((, ^ 0, 

 a , b', c 



used to denote the lineo-linoar function 

 (a x + 6 y + e 



which include* 



used to denote the quadric function 



( o, *, g ) (x, y, i 

 I * */ 



The lot preceding notation is an instance of a tymmttrica! matrix : 

 the term* inr, itete tymmttrical, already explained with respect to 

 determinant, apply also to matrices. 



RBSCLTAMT. If there be a system of equations between the 

 lame number of unknown quantities (it is assumed that the several 

 equations are of the form u = o, where u is a rational and integral 

 homogeneous function), then the function of the coefficients which 

 equated to zero expresses the result of the elimination of the un- 

 known quantities from the several equations, or (what is the same 

 thing) gives the condition for the existence of a set of values satis- 

 fying the equations simultaneously is the Rttultant of the equations, 

 or of the functions which are thereby put equal to zero. In the 

 case of two (non-homogeneous) equations involving a single unknown 

 quantity, we may say more briefly that the resultant is the func- 

 tion which equated to zero gives the condition for the existence of a 

 common root. In the particular case of a system of linear equations 

 between as many unknown quantities, the resultant is the determinant 

 formed with the coefficients of the equations. 



DISCRIMINANT. If [in a system of equations the functions equated 

 to zero are the derived functions of a single rational and integral homo- 

 geneous function with respect to each of the variables thereof, the 

 resultant of the system is said to be the discriminant of the single 

 function. The definition is easily made applicable to the case of a 

 non-homogeneous function, the functions equated to zero are here the 

 function itself and its derived functions with respect to each of the 

 several variables. For a single function, it may be said that the dis- 

 criminant is the function which equated to zero gives the condition for 

 a pair of equal roots of the equation obtained by putting the function 

 equal to zero. 



To fix the precise value of the discriminant of a given function, 

 it is assumed that the coefficient of some one selected term is + 1. 

 Thus, the discriminant of a x*+2 6 x y + c y' is a cb* : that of 

 ' 



In quadratic forms (in the theory of numbers) the expression 6' ae, 

 which is the determinant '* with the sign reversed, is called the 



determinant of Use form a x*+2bxy + cy*. And in like manner for 

 ternary quadratic forms, there is the same reversal of sign. It may be 

 said as a convenient definition, that the determinant is the discriminant 

 taken negatively. 



Putxcs. It frequently happens, in problems of elimination and in 

 other problems, that a given number of relations existing between 

 a system of quantities can only be completely expressed by means 

 of a greater number of equations. Thus, to take a very simple 

 instance, if the unknown quantities x, y, are to be eliminated between 

 the three equations a x+by=O, a'x+b'y=o,a"x+b"y=O: this im- 

 plies two relation. between the coefficients a, b, a', b', a",b" ; but these 

 relations cannot be completely expressed otherwise than by means 

 of the three equations aV-a'b=o, a!b"a"b'=o,a"b-ab" = o; for 

 taking any two of these equations, e.g. the first and second, these would 

 be satisfied by o' = 0,6' =o, which however do not satisfy the third 

 equation and are not a solution. Such a system of equations, or 

 generally the system of equations required for the complete expression 

 of the relations existing between a set of quantities (and which are in 

 fftmtral man numerous than the relations themselves) is said to be a 



RATIONAL AICD ISTKORAI, FUNCTIONS (dotation and Nomtnclalurc 

 of)- A rational and integral homogeneous function, such as the 

 function ax* + 2bxy + cy' is denoted by 



(*)(', V) 1 



*h* th coefficients are only indicated by tho asterisk, but ore not 

 expressed. A non-homogeneous rational and integral function is con- 

 owed as derived from a homogeneous function by putting one of the 

 variables thereof equal to unity, and is represented accordingly ; thus 



^ <)(*,. 



But it oftenproper to express the coefficients, and in regard! to this the 

 following dutincUuu is made, namely 



denotes ax* + 2bx y + ey* ; and in like manner (a.i.r.rf) (.r,y\* denotes 

 a x> + 3 i xy + 3 c * y* + d y*, &c., the numerical coefficients being those 

 of the successive powers of a binomial. But when such numerical 

 coefficients are not to be inserted, this is denoted by an arrowhead, or 

 other distinctive mark ; thus (a, 6, e)t(x, ) denotes a x*+bxy + cy*. 

 A rational and integral function of any order is termed a quant if, and a 

 function of the orders two, three, four, five, 4c., is termed a quadric, 

 cubic, qvartie, qirintic, ftc., respectively. The number of variable* 

 (the function being homogeneous) is denoted by the words binary, 

 ternary, Ac. As a correlative term to coefficients, the variable* have 

 been termed faciatt*. A function which is linear in respect to 

 several distinct sets of variables separately is said to be tantii>artile : 

 or, when there are two sets only, linto-lincar. Thus a determinant is 

 a tantipartite function of the lines or of the columns; the function 

 arx' + bry'+c.r'y + dyy'uin Unco-linear function of (x ) y)and(' > y / ); 

 a notation for it is 



(a,b 



\*,V\ 



such as has been spoken of in regard to matrices. 



EMASAST. The development of an expression such as 



is naturally written under the form 



+ y (*) (x, y) > &, sO A > 



) /-, 



and the coefficients of the successive terms X", X*~V *< ftre raid to 

 be the emanants of the quantic (*) (x, y)". The coefficients of A.", or 

 0-th emanant, is the quantic itself, and the coefficient of /i", or ultimate 

 emanant, is the quantic with (x', y 1 ) in the place of (.r, y) ; but the 

 intermediate emanants arc functions of (x, y) and (x 1 , tf) homogeneous 

 in respect to the two sets separately. The coefficients may of course 

 be expressed thus, the emanant (first emanant) of (a, 6, c) (x, y)* is 

 (a, 6, c) (x,y) (x > ,yf), which stands for axxf + 6 (xy' + x 1 y) + cyy". 



LINEAR TRANSFORMATIONS. In this theory the variables of a function 

 are supposed to be respectively linear functions of a new set of variables, 

 so that the function is transformed into a similar function of these 

 new variables, with of course altered values of the coefficients, and the 

 question was to find the relations which existed between the original 

 and new coefficients and the coefficients of the linear equations. The 

 determinant composed of the coefficients of the linear equations is said 

 to be the modulus of transformation, and when this determinant is 

 unity the transformation is said to be unimotlular. It was observed 

 that a certain function of the coefficients, namely, the discriminant, 

 possessed a remarkable property, found afterwards to belong to it as 

 one of a class of functions called originally hyprrdctcrminants, but now 

 invariants, and it was in this manner that the problem of linear trans- 

 formation led to the general theory of covariants. 



INVARIANT. An invariant is a function of the coefficients of a 

 rational and integral homogeneous function or quantic, the characteristic 

 property whereof is as follows: namely, if a linear transformation is 

 effected on the quantic, then the new value of the invariant is to a 

 factor prtt equal to the original value ; the factor in question (or 

 quotient of the two values) being a power of the modulus of transfor- 

 mation, and the two values being thus equal when the transformation 

 is unimodular. The easiest example is afforded by the quadric function 

 (a, b,c) (x, y)* ; effecting upon it a linear transformation, suppose that 

 we have identically 



then it may be easily verified that a'c'-i'* = (o8-/Sy)(oo-l). The 

 invariant a e 4* is however in this case nothing else than the dis- 

 criminant ; as another example take the quartic (a, b, e, d, e) (x, >/), 

 for which a e 4 b d + 8 c*, a e e a d* b*d + 2 6 e a c 3 are functions 

 possessed of the like property of remaining to a factor prit unaltered 

 by the transformation, and are consequently invariants ; it may be added 

 that calling them I, J, respectively, the discriminant is hero = l' 27 J 1 , 

 a rational and integral function of invariants of a lower degree. 



COVARIANT. Instead of a function of tho coefficients only, we may 

 have a function of the coefficient*) and variables, possessed of the like 

 property of remaining unaltered to a factor prit by the linear transfor- 

 mation : such function is termed a covariant. Thus, a covariant (the 

 Hessian) of the quartic (a, 6, c, d, e) (.r, y) is 



(ac-b\2ad-2bc,ae + 2bd Sc\2be-2ed,ee-d t )(x,yy. 



The quantic itself is one of its own covariants. The term covariant 

 may be used in contradistinction to, or as including, invariant. The 

 terms invariant, covariant, have been explained in reference to the 

 simple case of a single quantic containing but one set of variables, but 

 they apply equally to the case of a system of quantics, and to quantics 

 which are homogeneous functions of two or more distinct seU of 

 variables. There is one case which it is proper to mention : if in con- 



