64 Royal Society. 



there are many theorems which are common to all the orders. The 

 following theorem is then proposed as having that character. 

 If the terms of a series be 



1, or (l+w)°, (1-Fm)', (i+w) 2 . . . #c. (l+n) p , 



the 1st + 1) terms of (1 +nf +1 



the 1st (j) + l) terms of (l+w) p+2 



the 1st (p + 1) terms of (l+w/ +3 



+ &c. &c. 



(if p and w be both not less than 1), any number will be the sum of 

 not exceeding (pn + \) terms of the series; in other words, pn\-\ 

 is the notation-limit of this series. 



It is manifest that this series is of such a form, that by varying 

 n and p, it is capable of expressing every possible arithmetical series, 

 also every possible geometrical series (each having 1 for the first 

 term) ; it will also express all the intermediate series of the success- 

 ive orders (to an indefinite extent), which exist between and con- 

 nect together by a regular gradation (as is well known) any such 

 arithmetical series with a geometric series, whose common ratio is 

 the 2nd term of both series. The theorem may be statedjwithout 

 the series thus : — 



If any geometric series (having 1 for its first term) and (1+w) 

 for its common ratio, be stayed at the (p + 1 )th term and discon- 

 tinued as a geometric series, but be continued from that term as an 

 arithmetic series of the pth order, by forming it with the pth differ- 

 ence as the constant difference, and the other differences (which 



will be x, x% x 3 , &c x p ). The resulting series will be 



the series stated in the theorem above, and any number may be 

 formed by not exceeding (pn + l) terms, that is (pn+l) will be 

 the notation-limit of the series; if p becomes indefinitely great, the 

 limit of the series is a geometrical series, and it would become capa- 

 ble of expressing any number according to a system of notation 

 whose base or local value would be (1 +n). 



The proof of the theorem seems to depend upon this, that the no- 

 tation-limit assigned by the theorem is actually the notation-limit 

 of all the geometric terms and one more, at least, while the geometric 

 terms alone fix the law of the series and ascertain its elements (that 

 is, the first term and the successive differences) ; and as the com- 

 binations necessary to enable the series to fulfill its law, and carry 

 on the notation that belongs to it, are regulated by the series next 

 below it, viz. by the first rank of differences, while the supply of 

 new combinations (as the series advances and the number of terms 

 that may be used increases) is indicated by even a higher series than 

 itself, the new combinations are always greater, and at length inde- 

 finitely greater, than the number required. If therefore within the 

 range of those terms that ascertain and fix the law of the series the 

 law of its notation-limit can be obeyed, it must always (a fortiori) 

 be obeyed as the series proceeds to a greater number of terms and 

 to a variety of combinations increasing in a higher ratio ; and the 



