436 MR. A. B. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTJENT- 



And when the complete set of non-composite invariants, and the complete set of 

 non-composite identical covariants in each of the dependent variables are retained, the 

 independent non-composite mixed covariants consist only of Jacobians of the first 

 order of any one invariant, and each of the dependent variables in turn. 



Limitation on Number of Identical Covariants. 



75. The following gives the limitation on the number of proper identical covariants 

 in the original variable when the equation is a quartic ; and when there is given a 

 quantic, not an equation, of the fourth order the reservation mentioned in 67 is here 

 indicated. 



For the general equation the first few identical covariants in their present forms are: 



U 2 = ( - 1) tm" - (n - 2) u* 



U 3 = (u - I) 2 uW - 3 (n - 1) (n - 3) uu'u* + 2 (n - 2) (u - 3) u' 3 



U 4 = (n I) 3 uW 4 (n I) 9 (n 4) u 2 u'u a 3 (n I) 8 (n 3) wV 2 



+ 12 (n 1) (n 3) 2 ira' 2 t*" 6 (n 2) (n - 3) 2 u'* 



U 5 = (n 1)* ttV - 5 (n I) 3 (n 5) u s u'u* - 2 ( - I) 3 (5n 17) wV'u" 1 

 + 4 (n I) 2 (5n 2 36n + 67) wW" + 6 (n I) 2 (n 3) (5n 17) u*u'u" 2 



- 60 (n - 1) (n - 3) 3 uu' 3 u" + 24 (n - 2) (n - 3) 3 u' 6 . 



76. Taking first the case of the quartic equation 



we have as covariants of this equation 



U a = 3tm u 2w' 2 , 

 = 9ttV 



v - 27M s w" 8 



TJ 3 = 9ttV 9uu'u H 



from which 



U 4 + 3U 2 2 = 



whence from the differential equation 



U + + 3U 2 2 = 108 Q 3 wV - 27 Q,>i*. 



