310 ALGEBRAIC FORMS we find that they both satisfy the partial differential equation of Ex. gr. To find all the polars and co-polars of a|t| = F the second order =a 02U + 92w _ _ (% + tl(ya) + x+ + Zaayt2 + 3ax(ya)H* cxydy1 8*202/2 ~ ’ 2 2 z and the operation indicated occupies the same position in regard + Qax(ya)ayt1t2 + Za^yt* + (yaft* + 3(ya) ayt^t2 + S(ya)a yt^ + ayt 2. to the new forms that the O pi’ocess does in regard to polars. (bx + tx(yb) + t2byy=bl + 2bx(yb)t1 + 2bxbyt2 Gordan finds it convenient sometimes to denote the O operation ,; + (yb)2t21 + 2(yb)bytlt2 + b2ytl. »y | | ={xy) symbolically. In the same symbolism the new Multiplying and applying the formulas process 0 is denoted by x-^y^ + x^y^Xy. F0y°=4bl, Let 0(/) =—+ m.ndx{cy1 CxtfyzJ ’ a b K’y 0=tF^x-xw-j ^ x(yb) +• jalbl(ya), 5the operand being any form p™qy. ,t 1 n 1 2 . 3 Then 0C(2/6) = (a6)a^- (2/J) - , ■Fjr r)(APxby+ 'y r)c“xuxu‘y) ~.k nil Txffl' /^ — ki T_ y — k, and thence 0 ax (yb) ={ab) ax {yb)
F = a |’ ° ^Q l(ybf + albx(ya)(yb) + ~axbl(yaf, 0(ax) also JW/'ASJmby l/y — (ab)(ax)m ~ _ 1, 1 3 3 3 F V1 = iQaiby(yb) + ^a%bxby(ya) + ^Ib^yb) + -^ajoay(ya), and thence Olc(ax)mby =(ab)k(ax)m-1<:by~k, which should be compared with the result F° ’2 = ~alb2y + lalbxayby + ~axb]al, ^.k m n , m,-k-,n-k il axbL y = {ab) ax by . F l’ ° = iQal,(ya)(ybf + -^ixbx(yaf(ijb) + ^b2x(yaf, Moreover, it is easy to see that (xa)m{ya)n satisfies the 0 equation, and that F 1 £lk(xa)m(yb)n = (ab')k(xa)m~k(yb)n-k. l’ = laV)y(ya)(yb) + ^a%ay(ybf + axbxby(yaf + %ixb^y(ya)(:yb) We may catalogue results as follows :— ^■^ay(yaf, , ,*kam-kin-k G a~ b„ = (ab) by , x 1 2 1 _ k m-kj7i-k F 2= 1 2/’ i^xb^ya) + ^axnyby(yb) + Taxbxayby(ya) + -a^^yb) > (Jc m,-j *7i = a kam,-k,(by)
- n-k
T , “ ax (by) b x + 10^a2/(^a)> z^k nil t = (ab) , ,kam-k,(yb) ,*n-k 0 (yb) x F 2/ iQaxayby+ F)axbifl’yby + ^^bxay, ^(xa^by =abk,(xa)*m-kyhn-k, Ok(ax)mby = (ab) (ax) by , ,*n-k 0 (xa) (yb) =(ab) (xa) m-k,(yb) 0 (xa) (yb) =ab(xa) (yb) The important point is that we always obtain symbolic products when we operate with G or 0 on symbolic products. It is clear that we may write niiii , ,ka l a m-k-l,n-k-l = ab) b x y k l n Cl O a™(yb) =(-)k-k l m-k-l (yb)n-k-l tikOl(xa)mby =( - )lak(ab)l(xa)r‘ &kOl(xa)m(yb)n = (ab)kab(xa)r‘ (yb)71 foreshadowing an extension of the process of transvection which will be reached later. The co-polar processes, by which ax is converted to ax k(ya)k, and (xa)n to (xa)n kak, are representable by writing x -ty , x t for x x 11 2 + y1 1, 2 in ax, and x1 + ty2,x<2~ty1 for x1>x2 in (xa) ; for ax becomes {ax + t(ya)}n, and (xa)n becomes {(xa) + tay}n. The combined polar and co-polar process on a% can he obtained by writing x1 + (yt) ,x2 + yt for x1,x2, and expanding in powers of h,h‘, for then ax becomes ax-f t(ya) + Wiy, the nth power of which generates the functions ; similarly if the function be (xa)n we can substitute xi + ty ,X2--(ty) for X,X2, converting (xa) into (xa) + ti(ya) + ttfiy, and the nth power generates the polar functions. To find the polar and co-polar functions of the product Tiix=am,n m+n , xbx=px write Fj1 - fci - ya)%ky This, obviously, arises from px+n when we write x1 + (yt), X2 + yt for x.1 and x0J in p™+n, and take the co-factor of (‘^+, "V ^')tk1 Hki 2 bince m+n (Px + h(yp) + t‘iPy) = (ax + h(ya) + t2ay)™(bx + t^yb) + hby)n, ■' = coefficient of tkH^ in the product (ax + h(ya) + t2ay)m(bx + t^yb) + t2by)».
Ff0=%tx(yaf(ybf + ybx(ya)yb), a a a 1 3 Fr 3,1 y = iQ xby(yaf(yb)+Y(jax y(y )(yb)‘ +10M2/(z/«) 3 + jQb3ay(ya)'2(yb), 1 11 2 F 2= a b y iQ x y(ya)2 + laiP'yby(ya)(yb) + ^^(ybf + -bpayby(yaf
+ ^b3aiy(ya)(yb), F 3
y = iQaPIyby(ya) + ^^jby(yb) + —b^yb^ya) + f^b^yb),
F
—
2/
1=
F^tuPyby + ^b^flyby,
by(yaf(yb) + ^ay(ya)ybf,
F 2
|’ =^(yaf + %iyby(yaf(yb) + ^a?y(ya)(ybf,
F 3
|’ =Y^yb^yaf+Y)ayby(ya)(yb) + Y^ay(yb)2,
¥ i= a
y
F
' yby(ya) + -a3yby(yb),
r=46r Of any order, Jcl + k2 — k, there are k + 1 polars corresponding to the binary compositions of k. It will be remarked that the sum of the coefficients of the terms of the polar is always unity. A term, without its numerical coefficient, is termed a member of the polar. If we take a member of the polar F^k]1’ ko2 and put in it b = a, we obtain <-i^(yaf l; and this is exactly what we obtain on putting b = a in fJ,1’^2; hence, making this substitution, we find that the sum of the coefficient of the members is unity. What we may call the leading member of F^1’*2, where F = aq£q involves kx factors of the kind (ya) or (yb) ; say (ya)kii(yb)kn where k^ + k^^k^; and also k2 factors of the kind ay or by say a 1 l 2 ‘y b y' where k2l + k^ — k2. So the member may be written a™ kn k'21bx k"22 ^a^by^iyaf^yb^1'2. We may arrange the members (i.) in descending order as regards the exponent of ax ;
