56 



Trans. Acad. Sci. of St. Louis. 



of series may be derived by applying the property named, 



thus: 

 If a, b, c, d, . . . . represent any series of integers, 



positive or negative, the set of series in which the 7i^' 



term of any series is equal to the sum of the first n terms 



of the preceding series will be : 

 a b c d 



a a + b a + b + c a + b +c + d 



a 2a + h 3a + 2b + c 4a + 36+2c + d 

 a 3a + 6 Qa + Sb + c lOa + Qb + Sc + d 

 a ia+b lOo + 46 + c 20a + l0b + ic + d 



Table 31. 



Table 31 may be arranged diagrammatically as in 

 Table 32. 



10 

 1111 

 12 3 4 

 1 3 6 10 

 1 4 10 20 



a + 



10 



oil] 



12 3 

 13 6 

 1 4 10 



+ 



10 



11 



12 



13 



14 



c + 



1 



1 



1 



1 



1 



&+ 



d + 



Table 32. 



