171-1-(E7r-1)!7r1 a?2 an! 7r 2 ! ... P141 P242 ��� -1 hp, - hpg = is converted into where dlo = d a P,q-1 - dapg

vanish. From the above D p4 is an operator of order pq, but it is convenient for some purposes to obtain its expression in the form of a number of terms, each of which denotes pq successive linear operations: to accomplish this write d ars and note the general result exp (mlodlo+moldol +... +mp4dp4 +...) =exp Mp g dp 4+�� .

where the multiplications on the leftand right-hand sides of the equation are symbolic and unsymbolic respectively, provided that m P4, M P4 are quantities which satisfy the relation exp (M14+Moir+...+Mp4EpnP+...) =1+mic -Fmoif+...+mp,eng+...; where E, n are undetermined algebraic quantities. In the present particular case putting m 10 = 1 2, mot= v and m P4 =o otherwise M10t+M01n+...+Mpot P n 4 +... =log (1 +�t+vn) M P4 = (_)p+4 -1(p+g 1)!�p p 4; p!g! and the result is thus exp(Mdlo+vdol) = {�die+vdol- 2 (� 2 d2 +2�vd11+ v2d02)+...{ =1 +,D10+vD01+... +1.0v4Dp4+...; and thence p010+ v d01 - (� 2d 20+ 2 � vd 11 +v 2 d02) +��� = log (1+IuD10+PDc.1+...+�pv4Dp4+...).

(-) Dp4= P+4-1 ,w (p11 -1)! " 1 ? (p2 +g2- 1) ! +1,q p l !gll p2!g2! )?n -1 d" �" lrl!7r2!... d d

the last written relation having, in regard to each term on th right-hand side, to do with 17r successive linear operations. Recalling the formulae above which connect s P4 and a m , we see that dP4 and Dp q are in co-relation with these quantities respectively, and may be said to be operations which correspond to the partitions (pq), (10 P 01 4) respectively. We might conjecture from this observation that every partition is in correspondence with some operation; this is found to be the case, and it has been shown (loc. cit. p. 493) that the operation 1 1 d P? 41 d p1 42 ... (multiplication symbolic) ?r1! ?2,�..

corresponds to the partition

(p1g1' rl p2g2 n2 ...).

The partitions being taken as denoting symmetric functions we have complete correspondence between the algebras of quantity and operation, and from any algebraic formula we can at once write down an operation formula. This fact is of extreme importance in the theory of algebraic forms, and is easily representable whatever be the number of the systems of quantities.

We may remark the particular result (-) p + p q! d p4sp4 +Dp4(pg)+1; d P4 causes every other signle part function to vanish, and must cause any monomial function to vanish which does not comprise ,one of the partitions of the biweight pq amongst its parts.

Since dp4+(-)P+T1(p +q qi 1)!dd4, the solutions of the partial differential equation d P4 =o are the single bipart forms, omitting s P4 , and we have seen that the solutions of p4 = o are those monomial functions in which the part pq is absent.

One more relation is easily obtained, viz.

=d P 4 lodp+1,4 -holdp,4+1+...+(-)r+shrsdp+r,4+s+.. daP4 References For Symmetric Functions.-Albert Girard, In- -vention nouvelle en l'algebre (Amsterdam, 1629); Thomas Waring, Meditationes Algebraicae (London, 1782); Lagrange, de l'acad. de Berlin (1768); Meyer-Hirsch, Sammlung von Aufgaben aus der Theorie der algebraischen Gleichungen (Berlin, 1809); Serret, Cours d'algebre superieure, t. iii. (Paris, 1885); Unferdinger, Sitzungsber. d. Acad. d. Wissensch. i. Wien, Bd. lx. (Vienna, 1869); L. Schlafli, " Ueber die Resultante eines Systemes mehrerer algebraischen �leichungen," Vienna Transactions, t. iv. 1852; MacMahon, " Memoirs on a New Theory of Symmetric Functions," American 1 Phil. Trans., 1890, p. 490.

Journal of Mathematics, Baltimore, Md. 1888-1890; " Memoir on Symmetric Functions of Roots of Systems of Equations," Phil. Trans. 1890.

III. THE Theory Of Binary Forms A binary form of order n is a homogeneous polynomial of the nth degree in two variables. It may be written in the form n n-1 2 ax 1 +bx1 x2 +cx 1 x 2 + ...; or in the form n n n=1 n n-2 2 +(1)bx x2+ ?

1112 which Cayley denotes by (a, b, c, ...)(xi, x2)n (i),(2)��� being a notation for the successive binomial coefficients n, 2n (n-I),.... Other forms are n-1 n-2 2 ax +nbx x +n(n-i)cx x +..., 1121 2 the binomial coefficients C) being replaced by s!(e), and n 1, n-1 1 n-2 2 ax 1 +l i ox l 'x 2 + L ?cx 1 'x2+..., the special convenience of which will appear later. For present purposes the form will be written a0x 1 +(7)a1x1=1 x2+ C 2)o'2x12 x 2 +...+anx2, the notation adopted by German writers; the literal coefficients have a rule placed over them to distinguish them from umbral coefficients which are introduced almost at once. The coefficients a 01 a1, a2,..�an, n+I in number are arbitrary. If the form, sometimes termed a quantic, be equated to zero the n+I coefficients are equivalent to but n, since one can be made unity by division and the equation is to be regarded as one for the determination of the ratio of the variables.

If the variables of the quantic f(x i , x 2) be subjected to the linear transformation x1 = a12Et2, x2 = a21E1+a2252, E1, being new variables replacing x1, x 2 and the coefficients an, all, a 21, a22, termed the coefficients of substitution (or of transformation), being constants, we arrive at a transformed quantic f% 1tn n n-1 n-2 52) = a S +(1)a11 E 2 + (2)a2E1 E 2 +��� in the new variables which is of the same order as the original quantic; the new coefficients a, a, a'...a are linear functions 0 1 2 n of the original coefficients, and also linear functions of products, of the coefficients of substitution, of the nth degree.

By solving the equations of transformation we obtain rE1 = a22x1 - a12x1, r = - a21x1 + allx2, aua12 where r = I = anon-anon; a21 a22 r is termed the determinant of substitution or modulus of transformation; we assure x 1 , x 2 to be independents, so that r must differ from zero.

In the theory of forms we seek functions of the coefficients and variables of the original quantic which, save as to a power of the modulus of transformation, are equal to the like functions of the coefficients and variables of the transformed quantic. We may have such a function which does not involve the variables, viz.

F(a ' a ' a ,...a) =r A F(ao, a1, a2,���an), 0 1 2 n the function F(ao, al, a2,...an) is then said to be an invariant of the quantic gud linear transformation. If, however, F involve as well the variables, viz.

F(-1-1 -1 t a a 0, a l, a 2 ,... ;51, 2) = r F(ao, al, a2,...; xi, x2),

the function F(a 01 a 1, a 2 ,... x i, x 2) is said to be a covariant of the quantic. The expression "invariantive forms " includes both invariants and covariants, and frequently also other analogous forms which will be met with. Occasionally the word " invariants " includes covariants; when this is so it will be implied by the text. Invariantive forms will be found to be homogeneous functions alike of the coefficients and of the variables. Instead of a single quantic we may have several f(ao, a1, a2...; x1, x2), 4 (b o, b1, b2,...; x1, x2), ... which have different coefficients, the same variables, and are of the same or different degrees in the variables; we may transform them all by the same substitution, so that they become

_, _, _, _, _, _, f(a °, a, a 2 ,...; (b 0, b, b 2 ,...; 1, S2),....

If then we find

F ( a, a 1, a 2,...b 0, b, b 2 ,...,. .. �; S = r A F(a 0, 711, a2,���bo, b l, b 2,���9���; xl, x2), viz.

)- ( p+4-1 (p - - q -1)!dpq+ ?l -) 1)!D'1 DT2 p!g! ....