MATHEMATICS, RECENT TERMINOLOGY IN. 



MATICA. 



613 



junction with a quantic (>) (x,y,z . .)" wo consider a linear function 

 x + riy+(z + . . , the invariants of the system are functions of the 

 coefficients of the quantic, and of the coefficients , n, . . . of the linear 

 function ; and treating these as facients, the invariant is said to be a 

 contravariant of the given quantic. 



The foregoing definition gives the characteristic property of a co- 

 variant, but it does not directly show how the covariauts of a given 

 quantic are to be investigated. This is supplied as follows : For any 

 quantic with arbitrary coefficients, for example (a, b, c,d) (x, y)*, there 

 exist operators involving differentiations in respect to the coefficients, 

 tantamount to the operators x d f and yd, in respect to the variables ; 

 thus the operator orfj + 2bd c +3cdd is tantamount to yd,, and 

 3bd a + 2bd, +cdd is tantamount to xd f . Or what is the same 

 thing, denoting for shortness these operators by {yd, \i\xd,} 

 respectively, the quantic is reduced to zero by each of the operators 



iyd, j yd,, [xd r j xd,. Any function of the coefficients and 

 e variables which, in like manner with the quantic itself, is reduced 

 to zero by these two operators respectively, is said to be a covariant ; 

 or, if it contains the coefficients only, an invariant of the quantic. 

 That the two definitions lead to the same result is of course a theorem 

 to be proved. 



The leading coefficient of a covariant, say the coefficient of x m in any 

 covariant of a binary quantic (*) (x, y) m , possesses the property of being 

 reduced to zero by the operator j y d, j , and has been termed a 

 peninrariant, but a more appropriate term is leminvariant. An in- 

 variant is a function of a given degree in the coefficients, and a covariant 

 is a function of a given degree in the coefficients and order in the 

 variables, and they may be and are designated accordingly ; thus, the 

 above-mentioned invariants I, j of a binary quartic are called respect- 

 ively the quadrinvariant and the culrinvariant, and the covariant of the 

 same quartic is termed the quadricovariant, or if the distinction were 

 required it would be termed the quadricoi-ariant quartic. In these 

 cases the designations are sufficient, but it is to be noticed that in general 

 there is more than one invariant or covariant of the same degree or of 

 the same degree and order, and that any such designation is only a 

 generic, not a specific, name. An invariant or covariant may also be 

 designated by a name referring to the mode of generation for example, 

 the discriminant. The name catalectlcant denotes a certain invariant 

 of a binary quantic of an even order : namely, it is a determinant, 

 which, for the above-mentioned quartic function, is 



a, b, c 

 b,c,d 

 c,d,e 



(being in this particular case the cubinvariant), and the name canoni- 

 sant denotes a certain covariant of a binary quantic of an odd order, 

 namely, it is a determinant the terms whereof are linear functions of 

 the coefficients, and which for the cubic (a, b, c, d) (x, y) 3 is 



I ax + by, bx + ey 

 bx + ey, ex + dy 



(being for the particular case the Hessian or quadricovariant). 



CANONICAL FORMS. A binary quantic of an odd order 2 m + 1 

 admits of being expressed as a sum of (m + 1) powers of linear functions, 

 for example, the cubic (a,6,c,d) (x, yf can be expressed in the form 

 (/ x + m y) 3 + (Cx + m'yf this ia the canonical form of a binary function 

 of an odd order. And there is in like manner a form (not admitting, 

 however, of a simple definition) which is taken as the canonical farm of 

 a binary quantic of an even order. The catalecticant and the canonisant 

 present themselves in the problem of the reduction of a binary quantic 

 to the canonical form. 



BEZOUTIC MATRIX . If v, w are any two binary quantics of the same 

 order m, v', w' are what V, w become when the variables (x, y) of 

 the two quantics are changed into (x 1 , y') ; then (v w* v w^-f-fa y'xfy) 

 ia a rational and integral homogeneous function of the degree m 1 in 

 each of the two sets (x, y), (a?,*/), and the coefficients taken in their 

 natural square arrangement constitute the Bezoutic matrix. The 

 determinant formed out of this matrix is in fact the resultant of the 

 two functions, or equated to zero it is the equation obtained by the 

 elimination of the variables from the two equations v=0, w=0. If 

 v, w are the derived functions of one and the same binary quantic of 

 the order m, then the corresponding matrix, being of course of the 

 ord^r (m - 2), is the Bezoutoidal matrix, and the determinant is then 

 the discriminant of the single quantic. 



It would be too long to explain the allied terms Bezoutiant, Cobe- 

 zoutiants, Bezoutoid, Cobezoutoids. 



TACTIXVABIANT, RECU'ROCANT. A definition in the language of 

 analytical geometry will be the most easily intelligible, and it can readily 

 be converted into an analytical form and made applicable to any 

 number of variables. The function of the coefficients which equated 

 to zero expresses that the two curves u = 0, v = 0, touch each other, is 

 an invariant, namely, it is the tactinvariant of the two functions 

 tJ, v. And in particular, if, instead of the curve v=0, we have the 

 line {.e + T)y + Az = 0, then the function which equated to zero expresses 

 that this line touches the curve u = 0, ia a contravariant, namely, it is 

 the rtfipr'jcant of the function v. 



FUNCTIONAL DETERMINANT, JACOBIAN, HESSIAN. If v, w be quantics, 

 then the determinant 



is the functional determinant, or Jacobian, of the quantics v, w. . . And 

 if v, w, . . are the derived functions of d,v, d,v, . . of one and the same 

 quantic u, then the determinant in question is the Hessian of the 

 single quantic : the Hessian is in fact to the Jacobian what the dis- 

 criminant is to the resultant. 



CONCOMITANT, COQBEDIENT, CONTRAGREDIENT. The theory of linear 

 transformations has been considered from a different point of view ; 

 instead of the variables of a function being put equal to linear functions 

 of a new set of variables, they are considered as being replaced by a new 

 set of variables, linear functions of the original variables. Two sets of 

 variables may be so related that when the first set is thus replaced by 

 a set of linear functions of themselves, the second set is also replaced 

 by a set of linear functions of themselves, the co-efficients of the two 

 sets of linear functions being related together in a definite manner ; 

 this is concomitance, or rather it is (what is alone here spoken of) simple 

 concomitance. The two most important kinds of concomitance are, 

 1st. Cogrediency, that is, when the substitution on the second set of 

 variables is identical with that upon the first set; 2nd. Contmgrediency , 

 that is, when the substitution on the second set of variables is the 

 inverse or reciprocal one to that on the first set ; it will make the 

 notion of contragrediency clearer to remark that if the variables x,y, . . 

 and {, rj, . . . are contragredients, then d,tj ... (which are linear functions 

 of x, y . . ) and ', y 1 . . . (which are linear functions of {,?)..) are so 

 related that ^x! + -n'y' + .. is =fx + riy+ ... It was from the con- 



... 



sideration of contragredient variables that the notion of a contravariaut 

 was first derived, but as above remarked, the notion is really included 

 in that of a covariant. 



COMBINANT. A combinant is a covariant (or invariant) of a set of 

 quautics of the same order, which, besides being a covariaut in the 

 ordinary sense of the word, is, so to speak, a covariant quoad the 

 system, that is, it remains to a factor prcs unaltered, when the 

 quantics of the system are replaced by linear functions of themselves; 

 the factor in question being a power of the determinant formed with 



is merely changed into (Ap /tr) 2 (a </ 2 b b' + c a'), and it is therefore a 

 combinant. It would appear that the notion of a combinant might be 

 extended to the case of a system of quantics not of the same order, 

 and that the resultant of the system of quantics could be brought 

 under the extended definition of a combinant, but this is a point 

 which has not been considered. 



The principal text-books on the foregoing subjects, are on deter- 

 minants : Spottiswoode's ' Elementary Theorems relating to Deter- 

 minants,' 4to, London, 1851 ; Brioschi, ' Teorica dei Determinant!,' 

 4to, Pavia, 1854, translated into French by Combescure and into 

 German by Schellbach ; Baltzer, ' Ueber die Determinanten,' 8vo, 

 Leipzig, 1857 (especially valuable for its references to the original 

 sources). On elimination : Faa de Bruno, ' Theorie ge'ne'rale de 

 1'elimiuation,' 8vo, Paris, 1859. And extending to nearly all the 

 subjects : Salmon, ' Lessons introductory to the modern higher Algebra,' 

 8vo, Dublin, 1859. The memoirs on the different subjects are very 

 numerous, and it was not thought expedient to give a list of them. 



MATICA or MATICO, Medical Properties of. This name is applied 

 to an astringent plant brought from Peru, where it has long enjoyed a 

 high reputation for its styptic properties. Doubts exist as to the 

 botanical origin of the plant, some ascribing it to .1 Labiate plant, 

 resembling a phlomis, while others refer it to a piperaceous plant, and 

 even assert that it is the piper anffustifolium (Ruiz et Pavon, tab. 64, a), 

 a native of Peru. By Miguel this plant is now referred to the genus 

 Artanthe, separated from Piper, and is called Artanthe elonyata. It is 

 a Steffensia, according to Kunth (in Linnsca ,' xiii. 609). The odour 

 of the leaves, somewhat resembling a mentha, and the large quantity 

 of volatile oil obtained from them, lend countenance to the opinion 

 of its being a labiate plant; while the alternate position of the 

 leaves in most of the specimens described entirely negatives this 

 notion. The probability is, that two distinct plants pass under the 

 name of Matico, which, though they have a distinct origin, have 

 similar properties. Frequent instances of this are found in Brazil, 

 where numerous plants are called caa-pcla; and several, reputed 

 antidotes to the bites of serpents, are all termed quaco: one of 

 these is the Eupatorium glutinosum, Kunth. The analysis of the leaves 

 seems to have been made on the piperaceous plant, which is stated 

 to yield a drink employed by the Indians to produce effects 

 similar to the bang obtained from the Cannabil Tndica. Matico has 

 been analysed by Dr. Hodges, who found it to contain, 1, chlorophylle ; 

 2, a soft dark-green rasin ; 3, a brown colouring matter ; 4, a yellow 

 colouring matter ; 5, gum, and nitrate of potash ; 6, a bitter principle, 

 Maticine ; 7, an aromatic volatile oil ; 8, salts ; 9, lignin. 



Cold water extracts, in about four hours, all the medicinal virtues of 

 the plant, and is an eligible means of administering it. A tincture is 

 also employed, and the powdered leaves are both given internally and 



