Royal Society, 63 



The memoir was illustrated by numerous drawings, and the gi- 

 gantic humerus of the Pelorosaurus and other bones were placed 

 before the Society. 



Feb. 21. — " On the Extension of the Principle of Fermat's The- 

 orem of the Polygonal Numbers to the higher orders of series whose 

 ultimate differences are constant. With a new Theorem proposed, 

 applicable to all the Orders." By Sir Frederick Pollock, Lord Chief 

 Baron, F.R.S. 



The object of this paper professes to be to ascertain whether the 

 principle of Fermat's theorem of the polygonal numbers may not be 

 extended to all orders of series whose ultimate differences are con- 

 stant. The polygonal numbers are all of the quadratic form, and 

 they have (according to Fermat's theorem) this property, that every 

 number is the sum of not exceeding, 3 terms of the triangular num- 

 bers, 4 of the square numbers, 5 of the pentagonal numbers, &c. 



It is stated in this paper that the series of the odd squares 1,9, 25, 

 4-9, &c. has a similar property, and that every number is the sum of 

 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, theocto- 

 hedral, 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 2rc 4 — n\ 



The paper next contains an extension of the theorem 8^ + 3 = 3 odd 

 squares, which was proved by Legenclre in his Theorie des Nombres; 

 every odd square equals 8 times a triangular No.+ l ; the theorem 

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

 4.3=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 x 

 represent any figurate number of the x th order, and F y any figurate 

 number of the ?/ th order, whether y be greater or less than x, 



8F a? +3 = 3, or (3 + 8), or (3 + 2 .8), or . . . (3 + ?>8),&c, 



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

 ther (provided p be greater than 2) — 



/>F^+3=3, or ($+p), or (3 + 2p), or (3+np), 



terms of a series whose general expression is piF v + l, and vice versa. 

 The author concludes from these considerations, that probably 



