MR. E. CUNNINGHAM ON THE NORMAL SERIES 



+ i*+- +( 1 -0 1 1 )* = 0, 



A precisely similar set of equations gives the relations connecting the constants 



The equations X just written determine uniquely a set of values for l p+l ...0 l l and 

 the coefficients .r .... 



The first of these equations gives 



m 



Since r,, is to be equal to : \ve must have 6\, +1 = /3 1( and these equations are 



\0/ 

 then satisfied. , , 



Similarly the first of the ?/ equations with ?/ = gives 2 p+} = p 2 ; and so for 



W 

 the other columns. 



The second of the equations X written more fully gives 



<'-0; = 0, (p r -pi)*, r + V r = 0, * = 2 . > w. 



These then determine x l save for its first element, in place of which a unique value 

 is given for p l . 



The third equation X in full gives 



x 1 " + a 1 V 1 = 0, r = (2, ..., n). 



Of these, the first gives l f -i, while the following determine sc 2 save for its first 

 element, but only in terms of the yet undetermined .r, 1 . 

 Of the next group, the first equation is 



This equation apparently involves the unknown .r, 1 explicitly, and also, in x 2 2 ...x 2 ", 

 implicitly. 



But the whole coefficient of x r l is 



U 



pS-p,/ 2 



'i n 



2 2 ' 



EO that ^-2 is given independently of o^ 1 . 



