482 MR. A. E. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



reduction of the identical covariants. They form, however, a complete series of 

 functions, that is, any invariant which is a function of 3 and derivatives of 3 can 

 be expressed algebraically in terms of the elements of the complete series. 



As in the preceding case of identical covariants, the 2rth derived invariant is of even 

 grade; and the invariant 3i2r can be replaced by i/> 3i2r (which is functionally equivalent 

 to it), where 



the coefficients a lf 2 , . . . , a r being given by the equation 



2r! 2r+5! 5! 

 2r-sl 2r-s+5\ s + 5l s! 



a. = 



The (2r + l)th derived invariant is of odd grade; and the simplest functional 

 equivalent is an invariant of the third degree in 3 given by 



^asr + i = e s^'s,r i(2r + 6) e' 3 t/ 3 , 2r . 



Similarly for the derived invariants which are functions of 0^ and its derivatives 

 alone. The simplified functional equivalent of 6 M]2 , is 



*,* = eve; 2 " - ftew-" + ... + (_ ly&e^e^-' +... + (_ info 



where 



2r! 2/*+2r-l! 2/t-l! 

 Pi 



2r sl 2/t + 2?- s 1! 2/t + 1! s! ' 

 and the corresponding simplified functional equivalent of @ M>2r + 1 is 



' r -\- ft , 



So far as regards the index-equation, the first of (63), for these functions, we at once 

 have, after substitution, the value 2/i + 2r for X in connexion with /v,or, and the value 

 3/x, + 2r + 1 for X in connexion with Vv^+i 



It has been assumed in both of these examples that the Jacobian is an invariant ; 

 it is interesting to verify this in connexion with the differential equationa 



135. Example III. The Jacobian of a derived invariant and the priminvariant. 



Let <f> be a derived invariant of 6 M , and therefore a function of 0^, and its diffe- 

 rential coefficients alone ; let p be the degree of <f> in M , and let v be its grade. Then 

 the index X of <f> is 



X = p.p -f v. 



