46 Trans. Acad. Sci. of St. Louis. 



Since the corresponding rows and columns in Talile 6 

 are identical, this table may be extended for negative 

 values of n towards the left precisely in the same way as 

 it was extended upwards in Table 10 for negative values 

 of r, thus leading to Table 11. 



Table 11. 



It will be noticed that the defining property of the num- 

 bers in Pascal's triangle— namely, that the n'^ term in 

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

 preceding series — ^where n represents the number of the 

 column, would be equally applicable if stated in terms of r, 

 where r represents the number of the row ; that is to say 

 Table 11 liecomes identical with Table 10, if in Table 11 

 each column is read from the top down and in Table 10 the 

 corresponding row is read from left to right. 



