24 MR. E. CUNNINGHAM ON THE NORMAL SERIES 



Further r 1 of the equations subsequent to these fail to determine the appropriate 

 element as above, on account of the quantities 1 h+s ] h+r being positive integers 

 for s = 1, 2...(r 1). These (1 1) equations are satisfied by putting terms 

 < ,* + *- l * +r a A+r,A+*( _ j , i) i n ^ i( wn ile of the other h + rl equations constants 

 a* +r '*(s = 1...A) can be found to satisfy h. Thus (rl) equations of condition are 

 found from this column, and therefore ^k(k 1) from this group of roots ; and so for 

 each group of roots. 



Assuming all the equations of condition to be satisfied, we have now the following 

 formal solution 



where ft is as follows : 



The square matrices about the diagonal of h, k... rows and columns respectively, 

 corresponding to the groups of 6*... which differ by integers, are of the form 



and all other elements, to the right of the diagonal and within the matrices of e^. 

 rows and columns about the diagonal corresponding to the groups of equal roots for # 

 are numerical constants, and all others to the left and right of the diagonal are zero. 

 Applying now the formula 



the solution is put in the form 



where in the last matrix all elements are zero that were zero in X i, and c i} is a 

 constant if ^-^ is a positive integer, but otherwise is a numerical multiple of t*-*. 



The expansion of the matrizant can be effected as on p. -22, with the result that in 

 the. expanded matrix the first h elements of the first row contain Iog(t/t ) to the 

 powers 0, 1...A-1 respectively, while the rest of the row is free from logarithms ; the 

 second row begins with zero, then unity, and the next (h- 2) elements contain log (/*) 

 to powers l...(h-2) respectively, and so on, the A th row being entirely free from 

 logarithms. In the (A+l) th row in the k places beginning with the diagonal term 

 occurs to powers 0, !...(&-!), and so on. 



