480 ME. A. B. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOT1ENT- 



the subslitution of this quantity, which must be a solution of the differential form- 

 equation, leads to the condition 



a. = 



s(n-s) 

 so that 



_ 2r! n 2r + a 1 ! n - s 1 ! 

 2r-sln- 2r -llsln-V. ' 



since is unity ; and this is true for s = 1, 2, . . . , r, so that ^ is determinate and 

 can replace U 2r . For instance, 



and this is functionally the same as U 4 reduced, for it is easy to verify that 



The index of <j>. y is evidently 2r (n 1), which should, therefore, be the value 

 of cr in the index-equation of <f>. 2r ; and the substitution of <f> 2r and comparison of 

 coefficients of ( 1)' a,u (>) u (2r '~ l) gives 



2<r -f n 2s 1 + n 2 (2r s) 1 = 0, 



which is true for all values of s, so that with this value of o- the index-equation is 

 satisfied. 



133. But when the fundamental covariant is to involve an odd derivative of u as 

 its highest, so that the grade is to be an odd integer, say 2?- + 1 (which is the case 

 with U^^, we may not take ^-^i to be of order in u so low as the second ; for, with 

 an arrangement of terms similar to that in <j> y) the last of them would be of the type 

 M (r V + 1) . When substitution takes place in the form-equation, this term gives rise to 

 a term {u (r) \* which will not occur in connexion with any other term in < 2r4 j, and, 

 therefore, for the satisfaction of the equation, would have a vanishing numerical 

 coefficient. The other numerical coefficients would similarly vanish, and the assumed 

 form of <f>y+i would be evanescent. 



The simplest form of ^+1 1K one which is of the third order in u, being a numerical 

 multiple of the Jacobian of u and <}> 2r ; we take as this form 



n - 2r 1 , 

 = M- 2 U,. 



