376 Trans. Acad. Sci. of St. Louts. 
A verification of these equations is shown in the fact 
hat 
Ott yy a. (3) 
This is equation (3). From equation (5) we have 
oi =aN’, 
which, with equation (3), gives the result 
at = aN. (6) 
In the former case, where a = 1 and » is an even num- 
ber, equation (3) gave 
ee ee 
9 ade 
— 
Equation (5) may be readily obtained by inspection, 
after the results of Table III are obtained, and it 1s 
readily seen that the values of @ and / in equations (4) 
may also be written 
a==Na—N+1 
I= Na + N—1. (7) 
In Table IV the group of terms having a summation 
value 2° is reproduced, and the summation values from 
a==17 to each succeeding term are given in the second 
column. In the third column the first factor is the num- 
ber of terms that have been added from and including 
a=17 to the various succeeding terms, and the second 
factor is the average value of the first term, 17, and sub- 
sequent terms. The result reached in Eq. (6) is shown 
in the final term of the series, where the final term / is 47. 
For any value of x the number of terms in the series 
when nv is an even number, is the same as in the series for 
the same value of x when » is an odd number and one unit 
greater in value. Table V gives the numerical values 
for the quantities represented in the above equations for 
x= 2 and «=3 for values of n from 2 to 13. 
It may be pointed out that when x is the square of 
