68 G. H. KNIBBS. 



and (57), the indices may have all integral values from 1 to n + 1 

 when n is even, and from 1 tow when nis odd, provided that the 

 coefficients /3, y etc., are suitably determined. That is to say the 

 series (72) will in all cases have 2m + 1 lines, m being the number 

 of coefficients, whether n be odd or even, when p, q, etc., are the 

 successive integers 1, 2, etc. 



We have already seen that when p = 1, A lt B u C 1 etc., are all 

 zero, and hence /3, y etc. may have any values whatever : it has 

 also been shewn that when n = 2, or n- 3, the indices 1, 2, and 3 

 are satisfied, provided that jS has in the former instance the value 

 4, 1 and in the latter 3. Further it has been demonstrated that 

 when n = 4 and n = 5, the integral indices extend to 5, /3 and y' 

 being s f- and -f- in the former, and (3 and y -ff and ff in the latter 

 case. Moreover it may also be readily verified that when n — Q a, 

 septimic function is satisfied, and only a septimic when w = 7. 



It may also be noted that all the equations in (72) are not 

 independent. For example 



K 2 _ (2+ l)\k 2 + (n-k) 2 } -2n 2 _ 1 

 X; ~ (3+ V)p? + {n-kfl -2n s ~ Yn 



that is to say the values A 3 , B ;5 , C 3 etc, are simply 2?aA 2 , 2wB 2r 

 2t2C 2 , etc. Again if p denote an even (par) number, and i the 

 odd (^mpar) number a unit greater than p, that is % — p + 1, then 

 we shall have 



K' = — . K =^n^-pn^k+^^)n^- 2 k 2 -...-pnk^ l + 2^ (74) 

 ■p+ 1 p+i 2 ! 



K: = J- . K> = ^r^n 1 - 1 /^!^^-^^ (75) 



that is Ki has the same number of terms as K p , and k is raised to 

 the same powers. We may divide this last equation (75) by n, 

 hence substituting p + 1 for i we have 



K -=^W) • K »-= ^- {P+ i^k + m?«- ... 



... - 2\ nkV ~ l + (P +l ) kv ■ • ( 75a ) 

 l /3' will be 2. 



L. < 73 > 



