294
ALGEBRAIC
FORMS
or, in words, we express a member of the kth polar of /. ^ as a sum of the polar itself, and the product of (xy) into a linear function of the members of the k - 1th polar of the first transvectant of /over p. We may similarly treat any member oK/.*)’}*,-1 x x th of the k - 1 polar of the first transvectant, so as to exhibit it as a sum of the polar itself and a product of (xy) into a linear s function of the members of the k — 2th polar of the second transa general formula for the polar of a product of two forms. The vectant, /(ab) ,n2am-2.11-2 f „ , expression fy4Xi S is called a member of the polar ; writing it Gs bx =(/,!) x we have altogether a series of & + 1 members, viz.— of/ over p ; we can continue the process so as finally to reach the GojGj.Ga, ...Gk; development G and s = {(/ tfO } J{(/ /01 H “1 G +(/ ,?!)2)/ “2+• • •+Mkck(f, i)k. rrx/-4=2(^-*) S Hence the theorem which states that any member of the kth polar Two members, GS,GS+1, are said to be adjacent, and we can prove of the product of two forms/, (f may be expanded in ascending that the difference between any two adjacent members contains powers of the determinant (xy), the cofactor of (xy)s being a the factor (ab)(xy) ; for numerical multiple of the k - sth polar of the sth transvectant of /over p. All of these transvectants are, as we shall show pre( n ~ ^S+l n =fy sSjk-s jS+l.k-'S-l Py -Jy 9y sently, covariants. ■m-s-1 s.n-k+s.k-s-l r„ir„„. . =ax aybx by {ab){xy); Ex. gr. Let F =/. 4=ax.bx] and let us exhibit the second whence also member of the second polar in the desired form, a ) J x(.^s ~ Gs+l) —fy'Py (ab){xy), (i)F|=(2)(l)F^ + (?)(1s)F^ + (i.)(2)^«, s 1 th or wherein /^'Py “ “ is a member of the k~l polar of F=/.^. Again, if s be less than t, Fy 4* cc^ctyb^) 5 Gs - Gj = Gs - Gs+1 + Gs+1 - Gs+2 + ... + Gj _ j - Gt; the second member is a^ftyb^by and and since the successive differences MqfiLybgby ^^P'X^xPy y d* Q'xft'ybx QjfCl^bjifoy) G G s“ s+l Gs+l-Gs+2”-Ge-i-Gt» involve, each of them, the factor (ab)(xy) ; so also does Gs - G^ pft'ofiyfcljly — CLybft) the difference between any two members of the polar. Also the cofactor th of (ab)(xy) in axbx(Gs - G^) must be a sum of members of = (xy)abfaxbx=jxy)f, p'f ; the Jc - 1 polar of F. therefore Moreover, since a&ybtyy=pf, the result. In this instance the term, involving the first power S of (xy), happens to be absent. In regard to the polar of a proand duct of n forms it is easy to establish that if k
expanding and comparing coefficients of X ^m+n m+n-k k '.1 /"mf n am-saSjU-k+sJc-s ) by x Py'-= ^ s J.k- sj x i/ x ... m . ,n , or writing a =/, b =f,
^7,A = i72-fn = nT
S
we may write F
y~ coGo + ciGi+ "• + ckGk where CQ + CJ +...+Ci=l, and G= c « ( o + ci + • • ■ d- C*)G« 5 we obtain, by subtraction, which proves that the difference between the complete polar and any one of its members contains the factor (ab)(xy), and that the cofactor thof Gs) is a linear function of the members of the k - I polar of F. Since ® A(Gs - G) = (jy^y _ S _ 1 +fy+1(p,y “S ” 2 + • • • +/y ~ ” ^{ab^xy), we find " Gs)^-(a&)(aJ2/) =c
1
c +c () 2 ofy t y + (. 0 l)fy t y + ••• + (c0 +C1 + ••• + cs-1)/^
-(.Cs + Cs + 1+ ■■■+Ck)flpy~S~l - ...-CkJy-1^; and we see that the cofactor of (xy) in F^ - Gs is a sum of terms obtained by writing (ab) for axbx, in the cofactor of (ab)(xy), in the expression of «A(F^ ~ Gs). Hence the cofactor of (xy) in F _G i/ s a linear function of the members of the k — 1th polar of 1 (ab)a™ , which is a covariant of F =/. (f, usually written 1 an( (/A) thi termed (see post) the first transvectant of /over p. The k - 1 polar will be written and
{O'.*)1}?-1.
GK/.0)1};-1 will denote one of its members. We thus have the relation
where
2i = k. As regards polars, with more than two sets of cogredient variables, the generating function is (ax + ay-- /j.az+ ...)n; and we can establish the theorem that every symbolic product involving several sets of cogredient variables can be expanded into a number of terms, each of which is a complete polar, multiplied by a product of powers of the determinant factors (xy), (xz), (yz), .... Transvection.—Certain covariants, termed transvectants, have been met with above. We have seen That (ab) is a simultaneous invariant of the two different linear forms/=aa;, p = bx, and we observe that (ab) is equivalent to the differential operation 8/ df dt CXi 0*2 0*2 0*1" The process is generalized by forming the function (m-k) (n-k)!/0/ 0(ft 0/ 00 m n! *! 0*2 0*2 cx1 J ’ where /, 0 are any two binary forms ; it is called the kth transvectant of/ over 0. It should be noted that the multiplication of operations is symbolic in the sense that the operation in the bracket is to be performed k times successively. The transvectant is denoted by (/» 0)*Thus if f=a™ 4=bx, / m 7N TO-l.w-l ^x) ab)etx bx , / to .nk r~i.kam-k.n-k Vaxb ) =(ab) b , x
x
x
from which it is evident that the kth transvectant is a simultaneous covariant of the two forms.
