DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 443 

 being z" 1 , z'~ l z, z'~*z*, are given by 



P, P\e-*dx, ffi W-'dx}*, 

 wliere is determined by the equation 



Between any three linearly independent integrals there subsists a homogeneous 

 quadratic relation. 



The Quotient-Equation for the Ciibic. 



85. By this is to be understood the differential equation satisfied by the quotient of 

 two solutions of (22). Since every solution of the fundamental equation implicitly 

 contains, in linear and homogeneous form, three arbitrary constants, such a quotient 

 will implicitly contain five (= 6 1) independent arbitrary constants ; and the 

 differential equation which it satisfies will therefore be of the fifth order. 



Let j and 2 be any two solutions and s their quotient, so that 



Then, by (22), we have 



= t< *. 



and, therefore, 



= Wja 1 " + Sites' 1 + Stt"^. 



When this equation is differentiated and substitution is made for "', it follows thut 



and another differentiation and substitution give 



When MI, tt'i, w 11 ! are eliminated between these three equations, we have 



_ = <>, ... (24) 

 - Ss'e, 4s" 1 , 6s" 



3s", 3* 1 



the equation required, evidently of the fifth order. 



3 L 2 



