96 BISHOP TERROT ON THE SUMS OF THE DIGITS OF NUMBERS. 



niimbers substituted for x are together equal to six, will have the same ulti- 

 mate sum. Thus, in the above series, the 1st and 7th terms, in which and 6 

 are respectively substituted for x, have the same sum ; so also the 2d and 6th, in 

 which 1 and 5 are substituted, and so on. 



Pkop. XVI. 



In the series whose general term is m.m + 1 .... m + r-l, if m + w - 1 be 

 substituted for m, the ultimate sum of digits will remain as before. 

 If to each factor we add a, the term becomes 



a + mY.a + '>n + \-x.a + m + 2 . . . a + m + r—1 



. . . +m.m + l.m + 2 .... m + r—1, 



where a is a factor of every term except the last. Let a=n-l then term 

 7n + n-l.tu + n, &c. =m.)n + i, &c. +s.n^l, whose ultimate sum = that of 



m.m + l.m + 2, &c. 



Taking the same general term, if m + m^ — n-r, m'^ =n^i-7>i + r—l. 



Therefore wi,.»j J +I.OT1 +2 . . . m^ +r — l = n — l~m + r — l 



X n—l—m+r—2 

 X &c. 

 X n — l—m 



In this product, n^l will enter as a factor into every term except the last, 

 which is OT . w+ 1 . . . m + r—i with the sign ± according as »• is even or odd. 



If 1- be even, the m^ and m^^ terms wiU have the same idtimate siun ; but 

 if r be odd, the sums will be complemental. 



All the terms from the w-r"' to the n-1"' must have n-1 for their sum ; be- 

 cause w-1 must manifestly be a factor in each of them. 



Ex. Let r=2. Series is 1.2, 2.3, &c. 



=2, 6, 12, 20, 30, 42, 56, 72, 90, 110 



Sums are, 2, 6, 3, 2, 3, 6, 2, 9, 9, 2, &c. 



Let r=3, or series 1.2.3, 2.3.4, &c. 



= 6, 24, 60, 120, 210, 336, 504, 720, 990 



Sums, 6, 6, 6, 3, 3, 3, 9, 9, 9, &c. 



In the 1st example, ;• being even =2, m + m,^— 10-2 = 8; 

 therefore the 1st and 7th, 2d and 6th sums ought to be identical. 



In the 2d, ?• being odd =3, ?m + »«, =10-3=7 ; 

 therefore the 1st and 6th, 2d and 5th, &c. sums are complemental. 



Prop. XVIII. — Series of Figurate Numbers. 

 ' If the series be m, «b^+J:,^!t^+}:p:l, &c., where each term is the m'" 



