DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 461 

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



= 



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



./'/' 



and the primitive of the equation (45) is 



A i A f i A -rS 



AO ' M* '"'r / A c\\ 



*= U + KiZ+BS ' ;.-.' ' ' ' (45 >' 



where the quantities A and B are constants. 



Similarly, in general, the nth of these functions is the ntic quotient-derivative, 

 which will be denoted by [s, z]. Then 



. r [>,4. = (46) 



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



and the primitive of the equation (46) is 



_ AO + A t z + Agg* + . . . -I- A,,! f i .g|x 



* * . Tk . -n _a . . * *~1 *. ^*U j t 



where the quantities A and B are constants.* The equation (46) is of order 2n-l, of 

 course non-linear, though it is of the first degree ; its primitive (46 1 ) involves effectively 

 2n-l arbitrary independent constants. 



111. In the case of the quadratic derivative, the primitive (44') of the equation (44), 

 obtained by equating the derivative to zero, is symmetrical qua function of the 

 variables in 5 and z. Regarded in this light, the variables in the equation may be 

 interchanged, so that the equation 



[>, 2 ] 2 = 

 implies the equation 



[z, 1 = 0, 



This result was given by CAPT. MxcMAHON in A note, unknown to me at the time of reading of this 

 memoir, in the ' Philosophical Magazine ' for June, 1887, p. 542. 



