The General Term Of A Seminvariant Of Degree 0, 0 And Weight W Will Be A A A Appb°Ob°1B°2...B°4 _ 0 1 2 P 0 1 2 Q P Q P Q Where Ep S =0, Eas=0 And Esp, A Es,=W.

1111 The Number Of Such Terms Is The Number Of Partitions Of W Into 0 0 Parts, The Part Magnitudes, In The Two Portions, Being Limited Not To Exceed P And Q Respectively. Denote This Number By (W; 0, P; 0'. Q). The Number Of Linearly Independent Seminvariants Of The Given Type Will Then Be Denoted By (W; 0, P; 0', Q) (W; 0, P; 0', Q); And Will Be Given By The Coefficient Of A E B E 'Z W In L Z 1 A. 1 Az.1 Az. ... 1 Az P. 1 B. 1 Bz. 1 Bz 2 ... 1 Bz4' That Is, By The Coefficient Of Z W In Zp '. 1 Zp 2. ... 1 Zp 0.1 Z 4 1.1 Z 2. ... 1 Z4 E, 1 Z. 1 Z 2.1 Z 3 .... 1 Z 0.1 Z 2.1 Z 3 .... 1 Ze'; Which Preserves Its Expression When 0 And P And 0 And Q Are Separately Or Simultaneously Interchanged.

Taking The First Generating Function, And Writing Az P, Bz4, 2 For A, B And Z Respectively, We Obtain The Coefficient Of Aobe'Zpo 0' 2W That Is Of A E B E 'Z �, In 1 Z 2 1 Azp. 1 Azp 2....1 A2 P 2.1 Az P . Q 1 The Unreduced Generating Function Which Enumerates The Covariants Of Degrees 0, 0' In The Coefficients And Order E In The Variables. Thus, For Two Linear Forms, P =Q = I, We Find 1 Z 2 1 Az. 1 Az I. 1 Bz. 1Bzl' The Positive Part Of Which Is 1 1 Az. 1 Bz. 1 Ab' Establishing The Ground Forms Of Degrees Order (I, O; I), (O, I; I), (I, I; O), Viz: The Linear Forms Themselves And Their Jacobian J Ab. Similarly, For A Linear And A Quadratic, P= I, Q= 2, And The Reduced Form Is Found To Be 1 A2B2Z2 1 Az. 1 Bz 2.1 Abz. 1 B. 1 A2B' Where The Denominator Factors Indicate The Forms Themselves, Their Jacobian, The Invariant Of The Quadratic And Their Resultant; Connected, As Shown By The Numerator, By A Syzygy Of Degreesorder (2, 2; 2).

The Complete Theory Of The Perpetuants Appertaining To Two Or More Forms Of Infinite Order Has Not Yet Been Established. For Two Forms The Seminvariants Of Degrees I, I Are Enumerated By 1 Z, And The Only One Which Is Reducible Is Ao 0 Of Weight Zero; 1 Hence The Perpetuants Of Degrees I, I Are Enumerated By 11 1 � Z 1Zz' And The Series Is Evidently A O B 1 Aibo, A 0 B 2 A B A2Bo, A O B 3 A L B 2 A 2 B 1 A3Bo, One For Each Of The Weights I, 2, 3,..Ad Infin. For The Degrees I, 2, The Asyzygetic Forms Are Enumerated By Z. 1 And The Actual Forms For The First Three Weights Are 1 Aobzo, (Ao B 1 A 1 B O) Bo, (A O B 2 A 1 2 0 Bo, Ao(B2, 3 A1B2 A2B1 A O (B L B 2 3B O B 3 ) A I (B 2 1 2B 0 B 2); Amongst These Forms Are Included All The Asyzygetic Forms Of Degrees 1, 1, Multiplied By Bo, And Also All The Perpetuants Of The Second Binary Form Multiplied By Ao; Hence We Have To Subtract From The 2 Generating Function 1Z And 1 Z Z2, And Obtain The Generating Function Of Perpetuants Of Degrees I, 2.

1 _ 1 _ Z 2 Z3 1 Z. 1 Z 2 1 Z 1 Z 2 I Z. 1 Z2' The First Perpetuant Is The Last Seminvariant Written, Viz.: A O (B O B 2 3B O B 3) A L (Bi 2B0B2), Or, In Partition Notation, Ao(21) B (1)A(2)B; And, In This Form, It Is At Once Seen To Satisfy The Partial Differential Equation. It Is Important To Notice That The Expression (0) A (0'Ls) B (01)A(0'18 1)B (812)A (0'18 2)B (Op). (0')B Denotes A Seminvariant, If 0, 0', Be Neither Of Them Unity, For, After Operation, The Terms Destroy One Another In Pairs: When 0, Must Be Taken To Denote Ao And So For 0'. In General It Is A Seminvariant Of Degrees 0, 0', And Weight 0 0' S; To This There Is An Exception, Viz., When 0=O, Or When 0'=O, The Corresponding Partial Degrees Are 1 And 1. When 0=0' =O, We Have The General Perpetuant Of Degrees I, I. There Is A Still More General Form Of Seminvariant; We May Have Instead Of 0, 0 Any Collections Of Nonunitary Integers Not Exceeding 0, 0 In Magnitude Respectively, (2 A2 3 A3 ...0 Ae)A(L S 2 G2 3 G3 ...0' Ge') B (12 A2 3 A3 ..0 Ab)A(1 S I 2 G2 3 G3 ...B Ge ) B (1 22A23A3 ...0 Ae) A(1822 G2 3 G3 ...0' Ge ') B () 8 (1 8 2 A2 3 A3 ...19'°) A(2 G2 3 G3 ...0' ' ') B, Is A Seminvariant; And Since These Forms Are Clearly Enumerated By 1 Z. 1 Z 2 .... 1 Z 0.1 Z 2.1 Z 3 .... 1 Ze An Expression Which Also Enumerates The Asyzygetic Seminvariants, We May Regard The Form, Written, As Denoting The General Form Of Asyzygetic Seminvariant; A Very Important Conclusion. For The Case In Hand, From The Simplest Perpetuant Of Degrees I, 2, We Derive The Perpetuants Of Weight W, Ao(21W 2)B A1(21"R 3) I A2(21" 4)B ..� Man 2(2)B, Ao(221W 4(B Al(221W 5)B A2 (221 " S)B ... Maw 4(22)B, Ao(231W 9B A, (221" 7)B A2(231W 8)B ... T Aw 6(23)B A Series Of 2(W 2) Or Of 2(W I) Forms According As W Is Even Or Uneven. Their Number For Any Weight W Is The Number Of Ways Of Composing W 3 With The Parts I, 2, And Thus The Generating Function Is Verified. We Cannot, By This Method, Easily Discuss The Perpetuants Of Degrees 2, 2, Because A Syzygy Presents Itself As Early As Weight 2. It Is Better Now To Proceed By The Method Of Stroh.

We Have The Symbolic Expression Of A /? Seminvariant.

2 Qj?(Alaal A2A2 ..� °E E 1 N 1 2 32 ��� ' 'Te,So')W Where A S A S Oi 1 2 = Si= ... = A 8J Si = Si = ... = B8; And A L A ... Ae T 1 T ... Rb=0.

And This Is True For 0 0' = Pare The Case Of The Single Observe That, If There Weight Of The Simplest

2 0 0 ' 0 " .. 1 I, As Can

To Obtain Information Petuants, Write

2 As Well As For Other Values Of 0 0' (Com Binary Form).

Be More Than Two Binary Forms, The Perpetuant Of Degrees 0, 0', 0",... Is Be Seen By Reasoning Of A Similar Kind. Concerning The Actual Forms Of The Per

Proceeding as we did in the case of the single binary form we find that for a given total degree 0+0', the condition which expresses reducibility is of total degree in the coefficients a and T; combining this with the knowledge of the generating function of asyzygetic forms of degrees 0, 0', we find that the perpetuants, of these degrees are enumerated by z26"'-11 -z. 1-z 2.1 -z 3 .... 1-z e. 1-z 2.1-z 3 .... 1-2e ..(1 +aex) =1 +Aix-+2x2-F... +Aexe .(1 +Te'x) =1 +Bix+B2x2+...+Bo'xe' Al+B1=0.

the condition is a1Ti=A1B1=0, which since A i =o, is really a condition of weight unity. For w = i the form is A i ai+Bib i, which we may write aob l -albo = ao(I) b -(I)abo; the remaining perpetuants, enumerated by z I - 2' have been set forth above.

For the case 8=1, 0' =2, the condition is a i r 1 72 = A032=0; and the simplest perpetuant, derived directly from the product A 1 B 21 is (I)a(2)b-(21)b; the remainder of those enumerated by z3 I z. 1-e may be represented by the form (1 X 1 +1) a (2 g 2 +1) b - (1A1)a(2g2+11)b+ ... t (22P11),; X 1 and 12 each assuming all integer (including zero) values. For the case 0=0' =2, the condition is a 1 a 2 T 1 T 2(a 1+ a2) (al +TO (al +T2) = -A2B 1 B 2 -A l A 2 B2 = 0.

To represent the simplest perpetuant, of weight 7, we may take as base either A2B 1 B 2 or A l A 2 B2, and since Ai+Bi =o the former is equivalent to A 2 ArB 2 and the latter to A 2 B i B2; so that we have,