SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 15 



In exactly the same way if X 2 denote the system derived from X by the change of 

 p _! into &p-i, and the equations 



Y X-X 2 



p- 1 ^ ~" ai on 



be formed, the quantities A,,_ 1 a.v+2 differ from y r only by the substitution of 6 p for 9 P \ 

 l p - 2 for 2 p - 2 ... and 6^ + 2 for 6*. The same operator A p _j applied to the equations Y 

 connects the columns y and z, and so on. 



Similarly, operators A p _ 3 ...A a may be defined. 



X 



Lastly, an operator Aj may be defined so that the equation AxX is -^-r where 



"i PI +p 



denotes the equations X with O-fp substituted for 8*. Then the equation Y r 

 will, when y^,... are replaced by aV,... and B p 2 ,..., 2 2 by P 1 ,..., 2 l , become the 

 equation A ? X p+r . 



Consider now the equation Z l ; it is independent of #/, Zi 1 ,..., and therefore reduces 

 to a simple numerical constant which must be zero (p. 13). 



But Y is a polynomial in 6*. It can therefore be only of the first degree, since 

 A P Y 2 is independent of P 3 ; is, in fact, the same as Z,, viz., zero, so that Y 2 does not 

 contain P 2 . It must then, like Z,, be only a constant, and must therefore vanish 

 identically, so that Y 2 = gives a further " equation of condition." 



Hence again X 3 cannot contain p l , and the operator A p _! connecting it with Y, 

 shows that it cannot contain l p -i. Thus X 3 again must be a vanishing constant, 

 giving another "equation of condition." 



Similarly, starting with the corresponding equation of the fourth column, we 

 find A P X 4 = 0, so that X 4 must be independent of 6 P . Also A^X, will vanish 

 identically, so that X 4 is independent of 6 l p - 1 , and similarly it is independent of 

 ffi p . a .... 



Thus if ej > 4, X 4 reduces to a mere constant which, as before, must vanish. 



The process may clearly be carried on as far as the e^' 1 column, so that the equations 

 Xi.-.X,^! all give equations of condition, as do also Y^.-Y^-js, Zj.-.Z^-a, etc. Starting 

 now from the second equation of the e/' 1 column, 



where ^ denotes the e^ column of TJ, it follows that the third equation from (e^ l) th 

 column must be a quadratic in 9 p -^~ l , independent of & 1 , {= <f> 2 (0p'~ l ) say}, and 

 such that 



Thus if 0/ 1 " 1 is one root of </> 2 = 0, 0/' is the other. 



