34 ME, E. CUNNINGHAM ON THE NORMAL SERIES 



If - = 2, A. l \. p - 2ti+l = k.ct p> and therefore if a p le ' ={= 0, the ej roots of this equation 

 are all different. If, however, - > 2, A*p_ 2ei+1 again is identically zero, and the 



necessary equations of condition are again satisfied. 



Proceeding thus we find, in fact, that if a p fl ^ 0, 6* kp) ..., 6' kp -k,^i', s = l.-.ei all 

 vanish and that &\- p - kei +i is the root of a binomial equation of degree e 1( whose roots 

 are all different, and so for the other divisors of k. If, however, a. p ' 1 = we have the 

 same equations of condition again necessary, viz., a/' 1 and a/'' 1 " 1 = 0, &c. 



Assuming, then, that a p lt > ^ 0, we find, without difficulty, that all the quantities 

 9* vanish for s = I...e 1; save those for which r is of the form k(pm/e 1 ) + l, so that 

 the exponential arising in the first ej rows involves only ?*'> and not t llk . 



The discussion of whether the solution of the subsidiary equation proceeds according 

 to powers of t l! ' 1 only in the first EJ columns will not be carried out in full here. It is 

 enough to know that, provided a,,' 1 ' 1 , a / / 1+1 ' <r '" ) " < - do not vanish, a subnormal form 

 certainly exists satisfying the equation. 



If, however, one or more of these quantities does vanish, and one of the consequent 

 equations of condition is not satisfied, we may, as on pp. 31-33, find a new integer /-, 

 such that the necessary conditions for the existence of the subnormal form are 

 satisfied. 



17. As a concluding example consider the system derived from the equation of third order and rank one 



Clsot 3 + Cl 3 iZ 2 



~ ~ .'/ T 

 which with 



gives 



if-' 



I " \ / Z 



.\-oss -a-2-2 O/ \~0ss -BO, 4/ \-fflgi - 20 O/ \-a 30 0; 



The characteristic equation is - p 3 - pa?, + a 33 = 0. 



We shall confine ourselves to the case in which this equation has three equal roots. These must all 

 then be zero, and a = 0, a 33 = 0. 



For the equation then to possess a normal solution we must have 001 = 0, a 32 = 0. Supposing these 

 conditions not satisfied, put z = t 3 ; then the equation becomes 



1 ON / O v 



! 1 I 



\ - a 3 > - 0-21 4/ 

 The subsidiary equations then become 



= 0, 3x. 2 s - OJxi* = ; 



i - (9,1 = 0, 3 3 3 - OJx-p - O^XT? = 0, - ^ 3 1 // - O'Sxi 3 - 3a 33 = 0. 



