lg MR. E. CUNNINGHAM ON THE NORMAL SERIES 



These quantities then being determined, consider now the equation X ,+ 3 , Y, 1+1 ... 

 and first it must be pointed out that the relations established between the equations 

 X Y... through the operators A (pp. 14-15), where the quantities 6 P , 0^... were 

 considered as independent, are still valid when P ^, &c., are determined as functions 



of 6 



The operator A, in the first place becomes replaced in the equation X, i+2 after that 



giving 0',-! b J ^ + Vi--VF?F- But ViX., vanishes owing to the choice of 



"p Up 



0Vi. so that the value of 6\^ being substituted in X e , +2 the operator A p may be still 

 applied to establish a relation with Y, I+ I. 

 We have further 



VA. = &-i(V). 



while A p _ 3 X,, +2 ... vanish identically because of the vanishing of Y,,- 2 , Y ei _,.... 

 Thus the equation X, i+2 is of the form 



in which p l and 1 p - l are to have their determined values, so that the equation may 

 be written 



The operation A p having been shown to be applicable to the equation in this form, 

 reasoning exactly as above shows that the equation for 2 P _ 2 reduces to 



so that the values of (J p - 2 associated with the roots o- 1; cr 2 are independent of the order 

 in which these roots are taken, and likewise the values of 6* p -. 2 ... will be unaltered 

 by a permutation of the same. The same may be shown in the same way of a 

 permutation of any other two consecutive roots, viz., that such permutation gives rise 

 to a corresponding permutation of the 0^_ 2 .... Identical reasoning leads to an 

 identical conclusion with regard to Q p -. a ...0 2 . 



Eventually we come to equations giving #,. When d p l ...0 2 l have all been found, 

 the equation 'K fi+p . l is of the form ^-i^ij^+x^i) = 0, where, as before, the 

 coefficient of 6* is not zero, so that 6* is determined like the rest ; while l 2 ...0i 1 are 

 found respectively from Y ei+p _ 2 .... 



All the 6's being now determined, if we pass to the equation X, 1+p and follow the 

 same argument that was required to prove the preceding equations independent of 

 Xj 1 ..., the coefficient of x is found to be the left-hand member of X. 1+p _i with (^ + 1) 

 substituted for 0^, i.e., it is simply $,^(0-1), which is not zero. Thus X. 1+p determines 

 the first of the undetermined elements x ____ 



Similarly in X, 1+p+1 the coefficient of x 2 l is 2< Ci _ 1 (o-i), so that by this x a l is given, 

 and so on for the succeeding equations in turn. 



