Royal Society. 61 



(l) + (2 + 3 + 4) + (5 + 6 + 7 + 8 + 9 + 10+ll + 12 + 13)...&c.= 



(l 2 + 3'H9' + 27- + (3")2) = 



a triangular number the base of which is the series 

 (1 + 3 + 9 + 27 + 3"). 



§ 2. CUBE NUMBERS. 

 If S=« 3 , the first term =n"+-(\—n). 



C. 



Every cube n 3 is the sum of an arithmetical progression of n 

 terms, the first term of which is unity, and the difference 2(n + 1). 



1 =P 



1 + 7 =23 



1 + 9 + 17 =33 



1 + 11 + 21+31 =43 



1 + 13 + 25+37 + 49 =53 



1 + 15 + 29+43 + 57 + 71 = 6 3 



1 + 17+33 + 49 + 65 + 81 + 97 =7 8 



D. 

 Every cube n 3 is the sum of an arithmetical progression of n 

 terms, the first term of which is the root n, and the difference 2». 



1 =P 



2 + 6 =23 



3 + 9 + 15 =33 



4+12 + 20 + 28 =43 



5+15 + 25 + 35 + 45 =5* 



6+18 + 30 + 42 + 54 + 66 =63 



7 + 21+35 + 49 + 63 + 77 + 91 ....=7 3 

 The last terms of these series are the alternate triangular num- 

 bers. If they be respectively divided by the first terms, tbe quo- 

 ticnts will be the series of odd numbers. 



E. 



Every cube n 8 is the sum of an arithmetical progression of n 

 terms, the first term of which is (n' i —n + 1), and the difference 2. 



1 =P 



3 + 5 =2 3 



7 + 9 + 11 =33 



13+15 + 17 + 19 =43 



21 + 23 + 25 + 27 + 29 =5' 



31+33 + 35 + 37 + 39 + 41 =6' 



43 + 45 + 47 + 49 + 51+53 + 55 =7 5 



