20 MR. E. CUNNINGHAM ON THE NORMAL SERIES 



and is therefore the same as if a 21 were zero. It is also independent of ft, 2 , so that Y 3 

 again is independent of a 21 . 



Thus until k is so large that Y* does not vanish independently of 0?, Y i+1 is 

 independent of a 21 , and therefore the same as was obtained in the foregoing, where a 21 



was neglected. 



Thus the insertion of a 21 in XP does not affect any of the equations Y^-.Y.^-* and 

 therefore the values of P 2 ...6* are independent of a 21 . 



But in the equation Y ei+7 ,_! the coefficient of <x 21 is the left-hand member of Y, 1+P _ 2 



with 0*+l for 6i* 



= (o- 2 -o- 3 )(cr 2 or 4 )... ^ 0. 



Thus a 21 can certainly be chosen so that the equation Y ei + p _! is satisfied. 



Having determined a 21 , it is at once seen from p. 19 that the following equations 

 now determine yS, yj... without ambiguity ; for since 0*...$? are independent of a 21 , 

 the coefficients of y,\ &c., are those found there whether a 21 be zero or not. 



In the same way for the third column, with a 32 , a 31 , taken into account, the equations 

 become 



= 0, 



and just as before the first equation in which : < 2 occurs with a non- vanishing coefficient 

 is the one following the equation from which -0* first does not vanish out identically, 

 viz., Z ti+;) _ 2 ; while 31 will occur first in the equation homologous to the Y equation 

 in which 21 first occurred, viz., Z, i+p _i ; in fact, in Z e|+p _ 2 , a 32 will occur multiplied by 

 the left-hand member of Z ei+p _ 3 with 0^+1 put for 0f, and in Z ei+p _ 2 , 31 will be 

 multiplied by the left-hand member of Y ei+p _ 2 with 0^+1 put for 0* : both these 

 factors are other than zero. 



Thus a :!2 can be chosen to satisfy Z ej4p _ 3 , and subsequently a 31 to satisfy Z, i+p _ 2! 

 while the preceding equations are quite independent of them both ; just as for y, then, 

 Zf 1 ... are given in succession without ambiguity. 



Treating the remainder of the first Ci columns in just the same way, all the elements 

 of these columns are found in succession, and the solution is complete as far as these 

 columns are concerned. 



The t- 2 columns associated with the next group of equal roots may be treated in 

 the same way, the singular equations being in this case the (e! + e 2 ) th of each set; 

 constants a ij will again be chosen in the matrix x to the right of the diagonal, 

 Ci + e 3 >: i> i + l, e l + e 2 >j>f l +l, to satisfy certain equations as above, and so for 

 each root in succession. 



Thus if the various equations for O p associated with the different groups of equal 

 roots of the characteristic equation have their roots all different, and the " equations 



