|g^ ON POLYGONAL NUMBERb. 



sponding value of s — r, making the series begin .with 0, 

 and increase by unity, until it ends with s ; 3d, these pre- 

 paratory steps being completed, take any value of ? — r in 

 the work, and find what numbers in the second column of 

 the table will produce the same when added together ; then 

 if the indices of these numbers, when added in like man- 

 ner, give the index of s- — r in the work, they are also the 

 jindices of the constituent polygons ; but if the sum of the 

 numbers taken from the table prove — to the given value 

 of s — r, while the sum of their indices is less than the 

 corresponding value of a in the work, the deficiency may 

 be Qiade up in the latter sum by the addition of units, be- 

 cause one is the index of in the table. 

 Example 1. Example Ut. Resolve 1 4 into pentagons. According to 



the directions given above, the work will st^nd thus : 



e=: J4; a— 14. 11 . 8 . 5 . 2 

 s — r = . 1.2.3.4 



Here the first value of 5 — rzzO resolves e into 14 units, 

 because in the table has 1 for its index ; and OX 14 r: 



— s r, and 1 x 14= 14 = a. The second value of 5 — r 



r: 1 resolves 14 into 9 pentagons of the first, and 1 of the 

 second order, for 1 in the table has 2 for its index, denoting 

 a polygon of the second order ; but 2 + 9=ll=a in the 

 work. The third value of 5 — r~2 resolves 14 into 4 pen- 

 tagons of the first and 2 of the second order, for 2 X 1 in 

 the table zz 2, the double of the index of which zz 4 and 

 4+1X4 = 8= «• The fourth value of * — rz=3 resolves 

 14 into 3 pentagons, namely, into 2 of the first and 1 of 

 the third order, for 3 in the table has 3 for its index, and 

 3^1X2 = 5 = a. The last value of s — r = 4 will npt 

 resolve 14, because 3 + 1=4 and the sum of their indices 

 re 3 + 2 = 5, which is greater than 2 or a in the work. 

 Examples. Example 2d. To resolve all the numbers from l6 to 24 



into tetragons or squares, which shall not in any case ex- 

 ceed 4 in number. It is evident fr-pm cor. 2, prop. 5, 

 and the last example, that all the values of a may be re- 

 jected in the present instance, which are greater than the 

 index of l6, namely 4, in the table to prop. 1, when this 



number 



