923 



not exceeding 10 odd squares. It is also stated, that a series con- 

 sisting of the 1st and every succeeding 3rd term of the triangular 

 series, viz. 1, 10, 28, 35, <&c., has a similar property ; and that every 

 number is the sum of not exceeding 11 terms of this last series, and 

 that this may be easily proved [it was proved in a former paper by 

 the same author]. The term "Notation-limit" is applied to the num- 

 ber which denotes the largest number of terms of a series necessary 

 to express any number; and the writer states that 5, 7, 9, 13, 21 are 

 respectively the notation-limits of the tetrahedral numbers, the octo- 

 hedrai, the cubical, the eicosahedral and the dodecahedral numbers; 

 that 19 is the notation-limit of the series of the 4th powers; that 

 11 is the notation-limit of the series of the triangular numbers 

 squared, viz. 1, 9, 36, 100, &c., and 31 the notation-limit of the series 

 1, 28, 153, &c. (the sum of the odd cubes), whose general expression 

 is 2?^*— 



The paper next contains an extension of the theorem 8?z -f 3 = 3 odd 

 squares, which was proved by Legendre in his Tkeorie des Nombres ; 

 every odd square equals 8 times a triangular No=-fl ; the theorem 

 therefore is-— 8 times any term in the ngurate series (1, 2, 3, 4, &c. ..) 

 -|-3=:3 terms of a series consisting of the next series, viz. (1, 3, 6, 

 10 . . &c.), multiplied by 8 with 1 prefixed, and also added to each 

 term. But it is stated that this theorem may be much extended; for 

 this is not only true of any two consecutive series, but generally if F.^ 

 represent any figurate number of the x^^ order, and any figurate 

 number of the t/^^ order, whether ?/ be greater or less than 



8F,H-3=3, or (3-f8), or (3 + 2.8), or . . . (3H- ?^8), &c., 

 terms of a series whose general expression is 8Fy + l ; and still fur- 

 ther (provided p be greater than 2) — 



p¥^ + 3=S, or (3+;?), or (3 + 2/?), or (S + ??p), 

 terms of a series whose general expression is pF^-j-l, and vice versa. 



The author concludes from these considerations, that probably 

 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 (1 +ny, (1 -^n)', (1 . . . &^c. (1 +71)^, 

 the 1st (jo + 1) terms of (i +?^)^''^^ 

 the 1st (p + l) terms of (l-\-ny^^ 

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

 + &c. &c. 



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

 not exceeding (p?i + l) terms of the series; in other words, jt??j + 1 

 is the notation-limit of this series. 



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

 n andjt?, 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- 



