16 MR E. CUNNINGHAM ON THE NORMAL SERIES 



Similarly, the fourth equation from the preceding column must give a cubic for 

 e p '~\ (fo = 0), such that 



Thus 0/'~ 2 , 0/ 1 " 1 , p tl are the roots of < 3 = 0. 



Eventually the first column gives an equation of degree e t for P (viz., X ei ), of which 

 the roots in any order are a possible set of values for p l ...0 p f '. Calling this equation 

 (f> (6} = 0, and denoting its roots in some assigned order by <r lt cr 2 , ...cr ] , let us consider 

 the values determined for l p -i... by taking p l = ov.. and P '' = <r tl . 



10. Again, as prior in order of simplicity, let the case in which the roots of <j>(0) 

 are all different be taken first. 



It has been shown above that the equation Y r , when x^... are substituted for y?... 

 and P \ 6 l p - 2 ... for P 2 , 2 p - 2 ..., becomes identical with A p _iX r+2 . 



Now, Y I _! is merely a polynomial in 6 P , independent of?/! 1 ... and 2 P - 1 ..., and 

 vanishing for P 2 = cr 2 , o- 3 ...cr ei . 



Let Y.,., = *_,(*/). 



Thus X ei+1 is linear in 6 l p ^ the coefficient of the same being ^,^(0^); the part 

 independent of 6 l p ^ contains only P , which is now a determinate quantity. 



If the roots of <j) ei (0 p ) are all unequal, ^.^^^O and 6 l p -^ is given uniquely ; and 

 similarly Y t] = is a linear equation for 2 p _i in terms of P 2 and 6 P , Z ] _! a linear 

 equation in 3 p -i, and so on. 



It is important now to consider whether the order in which the roots o- 1 ...o" ei are 

 taken is of significance in the solution ; that is, whether the value of ff'p-i associated 

 with a particular root a- k is the same whichever column of the dependent variables 

 this root is associated with, and whether a change in the order necessarily implies a 

 distinct solution, because, if so, the solution would appear to be by no means unique. 



The equation X, i+1 giving l p ^ is, we have seen, of the form <^ 1 (o- 1 )^ 1 p _i + />(o"i) = 0, 



where #(8) = 

 Thus 



Now fifa) is save for a constant factor 



(0-1-0-2) (o-j-o-a)...^-^), 

 so that ApX i+1 is 



= 0. 



But the equation Y I , which is independent of 0*,_,,... becomes, when p l (= a-,) is 



