ON POLYGONAL NUMBERS. 



number is considered separately; consequently, the cor- 

 responding values of a in all the remaining numbers, 

 17, 18 ... 24 may also be rejected, and the collateral va- 

 lues of 5 — r placed under the last series; which being done, 

 the work will stand thus. 



£ rr 16; a r= 4 . 2 . 



167 



e = 17; 



a — 5 , 3 . I 







e = 18; 



a = 6 . 4 . 2 . 







e:=: 19; 



a rz 7 . 5 . 3 . 1 







« =r 20; 



a =: 8 . 6 . 4 . 2 . 







€ - 21; 



a =r 9 . 7 . 5 . 3 . 1 







^ = 22 ; 



a =10 .8.6.4.2 



. 





<? r= 23 ; 



a =11 .9.7.5.3 



. 1 





e — 24; 



a =12 .10 . $ . 6 . 4 



. 2 



. 



r — s — 



6 . 7 .8.9 .10 



.11 



12 



Here the first index of l6 = 4, and*— r=6; therefore 



16 = a square of the fourth order by table; first index of 



17 =5 = 4+1; s^r~Q=6~^0; first index of 18 =: 

 6 = 44-1 + 1; * — /• = 6 + + 0: first index of 19 = 7 

 = 4 + 1+1+1; s — r=6 + + + 0; hence 19 is 

 resolved into 4 squares, 18 into 3, and 17 into 2: again, 

 second index of 20 = 6 = 4+ 2, s — r=7 = 6+l; 

 therefore, 20 is resolved into two squares, namely, one of 

 the second and one of the fourth order ; second index of 

 21 =7 = 4 + 2 + 1, 5— r = 7=6+l + 0; second 

 index of 22 = 8=4 + 2 +1 + 1, 5 — r = 7=:6+l + 

 0+0; therefore 22 is resolved into 4 squares, and 21 into 

 3 : again, second index of 23 = 9 = 3 + 3 + 2 + 1, s — r 

 = 7 = 3 + 3+ 1 +0; that is, 23 is resolved into 4 

 squares : lastly, third index of 24 = 8 = 4 + 2 + 2 ; 

 s — r=8 = 6+l + l; that is, 24 is resolved into 3 

 squares. 



In the same manner any other number may be resolved 

 into polygons of any denomination m, so that the number 

 of these polygons shall not exceed m, denoting the deno- 

 mination. 



II, 



