2 8 ME. E. CUNNINGHAM ON THE NORMAL SERIES* 



then give ^...yV-i. Y, +A then fails to give y,\ but c 21 can be chosen to satisfy the 

 equation and y? may be taken zero. 



The following equations then give the remaining elements in succession. 



This leaves the equation Y, in general unsatisfied, and a further equation of 

 condition is therefore necessary. 



Similarly, if 0,"-0, 8 = fi (an integer) of the equations Z, we can, by proper choice 

 O f C 32( r _ ,...2), satisfy the p-l following that which determines 6^, viz., Z,-^ ; 

 and just as a proper choice of the constants o r 21 enabled us to satisfy Y g - p+1 ...Y q -i, the 

 constants c r 31 can be chosen to satisfy Z f _f+i...Z 2 -i. 



Thus two equations, Z,, p , Z g , are left unsatisfied in general. The two remaining 

 constants, c, 32 and c?\ are utilised to satisfy the equations Z, +(1 , Z ?+A+ ^, in which the 

 coefficients of z* and z\ +li vanish respectively. To do this the terms c 31 ^" and c**"*' 

 are inserted in the third column of Xi- 



If then the indices O^.-.B,' be equal, or differ from one another by integers, exactly 

 similar treatment applies for each of the first I columns of 77, the i ih column furnishing 

 (i 1) equations of condition. 



For the (Z+l) tlh column, however, 1 r -0 1 ' +1 , (r^h) is not equal to zero or a positive 

 integer. Thus h w k (&i +l + m) does not vanish for any value of m, and the Ip constants 

 c r l+l ''(r =p...l, s = l...l) can be determined to satisfy the Ip equations between 

 Uq-,,, and Uj+1, U denoting an equation of the l+l" 1 column, and, in particular, ~U g -i p 

 being the equation determining 0/ +1 and U ?+1 determining u^. 



For the next column, however, l l+l 1 l+i may be a positive integer, X 1 say, so 

 that h <a k (# 1 ' t+2 + V) = 0, and the (</ + X 1 )"' equation, instead of determining the 

 appropriate element of >?, can only be satisfied at the expense of the 5 th , by making 

 the element above 9^ in ^ c' +2 ' m ^'. The </ th equation then becomes a further 

 equation of condition. Thus we shall obtain r 1 equations of condition from the 

 (/ + r) th column, associated with an index belonging to the second group of indices ^ 

 differing among themselves by positive integers ; and so on through all the indices as 

 far as #/'. 



A similar treatment is now applied to the columns (h+l)*,.(h+k), for which 

 0/' +1 = e r h+2 . . . = 6 r h * k (r = p. . .2) ; 6V l+1 is given as the root of an equation of degree k ; 

 and the minors of the determinants % whose diagonals are r h+l ...6 r h+k , have the 

 elements to the right of the diagonal suitably adjusted as above, while one equation 

 of condition is furnished in connection with every difference 1 * +r -r# 1 A+s , which is an 

 integer. 



Supposing these equations all satisfied, consider the expansion of the matrix 



--- + }> which is effected in just the same way as for p = 1 (p. 22). 



P + l Q 



If to = 2 p where Q r is a matrix made up simply of the diagonal terms r 1 ...0 r ", 

 the application of the equation 



