376 Trans. Acad. Sci. of St. Louis. 



A verification of these equations is shown in the fact 

 that 



?-t±N=x\ (3) 



This is equation (3). From equation (5) we have 



x a =xN 2 , 



which, with equation (3), gives the result 



i+l^xST. (6) 



In the former case, where a = 1 and n is an even num- 

 ber, equation (3) gave 



2 



Equation (5) may be readily obtained by inspection, 

 after the results of Table III are obtained, and it is 

 readily seen that the values of a and I in equations (4) 

 may also be written 



a = Nx-~ N + l 



l = Nx + N — l. (7) 



In Table IV the group of terms having a summation 

 value 2 9 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 

 = 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 I is 47. 



For any value of x the number of terms in the series 

 when n is an even number, is the same as in the series for 

 the same value of x when n 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 x = 3 for values of n from 2 to 13. 



It may be pointed out that when x is the square of 



