A Study of the Properties of Integral Numbers. 379 
The sum of the series, or 625, will then be a times the 
square of the number of terms, as is shown in the former 
discussion. 
Any number which is in the second column of Table I 
ean be thus treated. 
TABLE V. 
n sg i eee l a N a l 
Z 45-2 1-1 3 9 ms 1 5) 
3 as ee rae 5) 27 3 7 13 
4 Wa 4242 7 81 9 1 ws 
5 qo 41 1 ak 243 9 19 35 
6 be 4 St 1 4 31 a seer + | 1 53 
7 8 1-8 94 2S 2,181: | ~ 27 5D 107 
8 fia ig ae ts ae sot 6,561 | 81 1 161 
9 512440 1 | AE 19,683 | 81 163° S25 
0 4 2 0761-32 1 le 59,049 | 243 1 | 485 
it | 2,048 | 32.} 38] -95 177,147 | 243 | 487 | 971 
12 | 4,096 | 64] 1 | 127 531,441 | 729 1 | 1457 
13 | 8,192 | 64 | 65 | 191 * 1,594,323 729 | 1459 | 2915 
The series having a sum 3° appears in Table V as com- 
puted from equations (4). The first and last terms are 
a==3 and I—7. This is one of the results of Nico- 
machus. 
It is therefore apparent that any integral number 
raised to any power can be represented as the sum of a 
consecutive series of odd numbers. 
In the discussion which followed the presentation of 
this paper, Dr. C. H. Danforth suggested that the numerl- 
cal values represented in Table I might be represented 
graphically, as is done in the diagrams at the top of 
Fig. 1. Starting with a small square at the upper left- 
hand corner, having unit area, a series of equal squares 
representing consecutive odd numbers are added in the 
three diagrams on the right. The added areas are in each 
