SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 25 



14. Returning now to the general case left out of consideration on p. 16, in which the 

 roots of <f>(0 p ) = are not all unequal, we suppose the roots of this equation arranged 

 in groups, of which the members are all equal. 



If o-j = or 2 = ... = ov, < ei _i (<TI) = ^-^o-a) = 0, since the roots of <j> ei -i(0) = are 

 o- 2 , ...<r r ; and again, <f> fi _ 2 (a 2 ), <ji n _ 8 (<7 8 )...^ (n _ r+1) (<r r _ 1 ) are all zero, where 

 $,,-2 = 0....<^> ei _ r+1 = are the equations for dp, 6 P ...6 P . 



The equation for 6 l p --i (p. 16) reduces therefore simply to t|>(cri) = and fails to 

 determine 6 l p -i ; but, o-j being already known, this must be merely an " equation of 

 condition " among the coefficients. 



Similarly in the equation X, I+2 the coefficient of 6 l p - 2 vanishes, and this equation 

 therefore is of the form 



-i, 1 = or _ 1 , = 0. 



Now the operator A p _! acting on this equation, since cri = cr 2 , gives the equation Y f|> 

 which, as has been seen (p. 17), is linear in 2 p - l , the coefficient being ^> ei _ 2 (cr 2 ). 



If r = 2, this does not vanish, and therefore X e|+2 is a quadratic for 0\,-i, of which, 

 owing to the relation through & p -i, 6 l p -i, 2 p -\ are the two roots. 



If, however, r>2, the equation Y Ei must become an equation of condition, since 

 < ti _ 2 (o- 2 ) = 0, and therefore also X ei+2 becomes independent of 6 1 p _ l and gives another 

 equation of condition. 



Carrying on this reasoning step by step, we find that X^+i...X, 1+r _j are all 

 independent of l p -i, 6 l p - 2 ..., while X fi+r is of degree r in O 1 ^ and independent of 

 O l p - 2 .... If any root of this equation be taken for ffip^ the equations Y t] ...Y ti+ ,._ a arc 

 independent of 2 p -i, 6 2 P -^..., while Y n+r _ 2 is an equation of degree r 1, which, since 

 it is derived from X, i+r by the operator A ;) _ 1; has for its roots the remaining roots 

 ofX e , +r . 



Choosing one of these for 6 2 p -i, Z e]+r _ 4 gives an equation of degree ? 2 whose roots 

 are the remaining, and so on. 



Similarly, if tr r+1 = ... = ov +s , 6 p - l r+l ...0 p - l r+:i are given as the roots of an equation 

 of degree s, and so for each group of equal roots cr. 



Consider now one such group with the values of ^ p _, r+I , ...0 p _ 1 r+ * obtained. 



Let the equations of which these are the roots be 



Then, if the roots of $ s be all unequal, say = TJ...T,, 



^-i(ri)M=0, ^- 2 (T 2 )^=0, .... 

 but 



^_ 1 (r,) = 0,/^l. 



The subsequent equations are then seen by the application of A 3 to be 

 VOL, CCV. A, E 



