HKIUVATIVES ASSOCIATED WITH UNKAR DIFKKRRXTIAL EQUATIONS. 471 

 Hence the equation satisfied by /i is 



. V 

 , V 



- 5/t"Q 3 - 



, 8ft 1 

 , 6/x" 



= 0- 



The quotient-derivative for the present case it will be culled the hyperquadratic 

 derivative is obtained by selecting from the left-hand side of the quotient-equation 

 the terms independent of Q 8 and Q' a and by writing * for p, 1 ; thus it is 



"', 3*", 3s 1 

 6s" 



and then 



= 



(53) 



is the quotient-equation when Q 2 is zero. But in that case 



u, = B, + B l2 + &2, 



where a = 2&, so that 



A + 



C + D* + 65 s 



(53'), 



where, from the point of view of (53), A, B, C, D, b are arbitrary constants, is the 

 primitive of (53), containing four independent arbitrary constants. 



122. Proceeding in this manner we obtain for similar linear equations of successive 

 order a series of derivatives in each of which the order of the highest differential 

 coefficient entering is an even integer ; and their form is indicated in the following 

 scheme, similar to that of 110. The hyperlinear derivative is obtained by forming 

 the indicated determinant from the first two elements of the first two rows ; the hyper- 

 quadratic derivative is similarly obtained from the first three elements of the three 

 rows after the first ; the hypercubic derivative similarly from the first four elements of 



