10 MB. E. CUNNINGHAM ON THE NORMAL SERIES 



The series for x will terminate if for any value of n 



y + (q-n)(p + n-l) = 0, 

 ? f if 



for some value of n. 



The series for y will similarly terminate if 



(-A + 2/-l) 2 -(4y + l) 

 vanish for some value of n. 



Both these are certainly satisfied if 



X = q and 4y + 1 = p 2 , 

 where p, q are any integers of which one is odd and the other even. 



6. We pass now to the case where the characteristic equation \a p + 1 -p\ = has its 

 roots not all unequal, and the analysis becomes a good deal more intricate with the 

 less simple canonical form of the matrix a p+1 as stated on p. 4. It will be remembered 

 that the numbers e 1; e 2 ... there used are the powers of (p^-p) in the elementary 

 divisors of | a p+1 -p \ with respect to the root p l of this equation of multiplicity I In 

 the case of the system obtained on p. 4 from a single equation of order n, we may 

 prove that ^ = I, e 2 = e 3 = . . . = 0. 



For the matrix (a p+1 p) is of the form 



The minor of the quantity " k" in the determinant of this matrix is simply 

 ( p}"~\ Thus the elementary divisors are certainly merely unity with respect to any 

 non-zero roots. 



If there be a multiple zero root, however, since the minor of "a" is unity, the 

 elementary divisors with respect to this root are all simply p u . 



Thus for such a system we have for each multiple root e 3 = e 3 . . . = ; so that in the 

 canonical form of a p+1 , if 



Pi = pi+i> a p+i' ' = 1 

 and if 



Such systems being by far the most important in practice and also considerably 

 simpler to work out, the full discussion will be restricted to systems of this type. It 

 may be pointed out that the most general system can be solved by means of the 

 solution of systems of this more restricted type, for from the general system 



