4GO MR. A. B. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



the three lines after the first two ; the first four elements of the four lines after the 

 first three ; and so on, we obtain the series of quotient-derivatives connected with the 

 linear equations of successive orders. 



The first* of these functions, viz., s 1 , is the linear quotient-derivative ; in conformity 

 with the notation used for the remaining functions, it will be denoted by [s, z]^ 

 Then 



0,21 = . . .'. . ;. .... ..... .,,, (43) 



is the differential equation satisfied by the quotient of two solutions of the equation 



dit 



dz = * 



and the primitive of the equation (43) is 



where A and B are constants. 



The second of these functions, viz. : 



A 



(43') 



s", 2s 1 

 s m , 35" 



is the quadratic quotient-derivative ( 87); it will be denoted by [s, z] 2 . Then 



0. *1 = (44) 



is the differential equation satisfied by the quotient of two solutions of the equation 



= 0, 



and the primitive of the equation (44) is 



' 



B 



(44'), 



where the coefficients A and B are constants. 

 The third of these functions, viz. : 



*"', 3s", 3*' 

 s' v , 4s" 1 , 6s" 

 s", 5s lT , 



This function is inserted merely to make the enumeration complete. 



