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 ALGEBRAIC If m^n there are ?i + l transvectants corresponding to the values 0, 1, 2,...n of & ; if k — 0 we have the product of the two forms, and for all values oi k>n the transvectants vanish. In general we may have any two forms 0i> 02» 0i> 02 being the umbrae, as usual, and for the kth transvectant we have (<£, a simultaneous covariant of the two forms. We may suppose 0x> 0x be any two covariants appertaining to a system, and the process of transvection supplies a means of proceeding from them to other covariants. The two forms a™, 6™, or p, may be identical; we then have the kth transvectant of a form over itself which may, or may not, vanish identically ; and, in the latter case, is a covariant of the'single form. It is obvious that, when k is uneven, the kth transvectant of a form over itself does vanish. We have seen that transvection is equivalent to the performance of partial differential operations upon the two forms, but, practically, we may regard the process as merely substituting («&)*, (00)* for fc Jc ,k ,k respectively in the symbolic product subjected to transvection. It is essentially an operation performed upon the product of two forms. If, then, we require the transvectants of the two forms f +f, 0 + /10', we take their product ftp + f' or U k(a™by')y=x'= (/. 0)fc 5 so also is the polar process, for since ,k m-kJc Jc_rn-kJc fy~ax by, Yy bx by, if we take the kth transvectant of fy over <^, regarding ylt y2 as the variables, / .k .kk, iJc m-k,n-k .dt. f yx y) = (ab) ax bx , or the kthth transvectant of the £** polars, in regard to y, this equal to the k transvectant of the forms. Moreover, the k transvectant (ab)ka™~kbx~k is derivable from the kth polar of a™, viz., a™~kay by substituting for y^ y2 the cogredient quantities b2, - b^ and multiplying by bx k. Frequently the forms to which the process is applied are, each of them, products of other forms. Thus suppose TOi,m2 ._„'ihon2 f—Q'X ^X > $ — ax Px > we find / w m,,mo n no (wi1 + + W2)®x bx ,ax 10Px J =,rw',,*)„’• ■ V? + »>,%!«,'*>”'' ‘CC/S? ' ’ + V? + '1; where, on the dexter, there is a term for every pair of umbrae, one taken from / and one from 0. The sum of the^ numerical coefficients of the members of the transvectant is unity. If we suppose the product written out as a product of + m2 + ft! + ft.2 factors ; the coefficient (m1 + m2)(ii1 +w2) enumerates the number of ways of picking out one factor from a^b^and one factor from a”1/^2 > the coefficient m-^n^ on the dexter of the above results shows the number of ways of picking out one factor from a™1 and one from /3”2; consequently the fraction {m1 + m.d){nx + n,-,2i which affects one term of the transvectant, denotes the probability of picking out the pair axpx when a random selection is made of one factor from a™'fc™2 and one factor from a”1/?”2. Similarly if

295

FORMS

J —^X ^X CX ",SX J ^ — ax Px fx ’"UX > 2m=m, H,n — n, the first transvectant will consist of sa terms, and any term, involving say the determinant factor («/3), will have a numerical coefficient which denotes the probability of a random selection of two factors, one each from / and 0, being ax^x. The sum of the coefficients is unity. Proceeding to the second transvectant each term is treated in a similar manner, with the result that a number of terms are obtained ; one such term is, say and the numerical coefficients denote the probability of two random selections of pairs of symbols yielding {ap)(ay), and so on. Hence, in general, the kth transvectant involves terms, each of which has k determinant factors, and a numerical coefficient which denotes the probability of such factors arising from a random selection. Hence the sum of the coefficients of the terms must be unity. Ex. gr. {axbx, cxdx)l = ^(ac)bxdx + ^-{ad)bxcx ~^{J}c)cixdx-1r ^{bd^jci^x ; {axbx, Cj/lx)2=^{{ac){bd) + {ad){bc)}. Ex. gr. We will find the fourth transvectant, of a binary quartic upon' itself, so as to obtain the invariant (a&)4 in terms of the roots of the quartic. Let = bx~{X — ~ a2a:2)(a;i~ aix)> 1 and observe that (aq - a^, xx - a^aq) = ar - as ; therefore (o-x, &x) is a sum of a number of terms, each of which involves a factor of the form ar — a*, and proceeding we find (Kcix, bx) equal to _

_a

S>(al ~ ai)>

the summation being for every permutation ijkl of the numbers 1, 2, 3, 4. There are 4 ! terms, but certain of them vanish. Those which survive correspond to the permutations in which each number is displaced. These are nine in number, and we can, finally, throw the result into the form ^ -a

2)(a3 - “i) + (al ~ as)(a2 ~ a4) d (al _ a4)(a2 _ as)} > a well-known expression of the simplest invariant ofofthe binai} quartic. It will be seen later that the coefficient a0 every c°variant of a binary form is a symmetric function of the differences of the roots of the forms. . ^, The various transvectants can be obtained by partial dinerentiai operations in which the independent variables are ax, bx, ... ax, px, .... Dropping, temporarily, the suffix x, we can see that (CC, an^j can be obtained by operating upon the product with (aa)

^k +

+

+m

dbdp ’

Disregarding the multiand multiplying by (m + m )(wi + 7l2) 1 2 th plier we obtain the k transvectant by k successive operations of this operator, the quantities operated upon being a (tx, b — bx, ... and not the determinant factors. In general the operator is the number of terms in the operator being sa, and k successive operations produce, to a factor pres, the kth transvectant. Ex. n gr. To find in this manner the second transvectant of (abfa -2bn-‘2 upon itself, we take the product {abf(cd)‘2an~‘2bn-2cn-2 n 2 d ~ and perform the operation {(ac)'dade^^'dadd^^dbdc ' ^’dbddJ when we obtain the result in the form of a linear function of the three forms :— n 2 (a&)2(ac)2(ce£)%n-4&"-2cn-4, 3 1 c? (ab f lcd f lac^ad)^^’ --^-'^n n z n n , {ab)cd)ac)(fid)a -% - c -*d -*. A very important particular case of transvection is that in which f.. • , = gx-, 0 is the original form, / a covariant of 0, and M is the product of determinant factors involved in /. Writing for convenience m-i, me, n /=Ma b ..., <P=g ,