[ 120 j 



XIY. Note on an Equation in Finite Differences. 

 By J. J. Sylvester*. 



I GAVE a great many years ago in this Magazine the in- 

 tegral of the equation in differences 



X 



which I obtained by observing that the equation could be 

 solved by supposing each u of an odd order to be equal to the 

 u of the order immediately superior, and also by supposing it 

 to be equal to the u of the order immediately inferior. The 

 upshot of the investigation expressed in the simplest language 

 was to furnish two particular integrals of which one gives rise 

 to the series 



i i i i 1.3 1.3 



w =l %== l u 2 -\ u z — \ Ui --—- u 5 =— j . . . ., 



the other 



! 9 2.4 2.4 2.4.6 



1.6 1 . o 1 . o . 



See also Boole's Finite Differences, 2nd Edition (edited by 

 Mr. Moulton), p. 235. 



Now let </>, a function of any letter t, be the generating 

 function of u x . Then, since 



xu x - (x - 2)u x _ 2 - u x _ y - 2u x _ 2 = 0, 

 we shall have 



(l-Of+(-l-20<^=C; 



and integrating we find 



( i_ i ( i f0 i (/)= cj^ A /|^, 



1 + t 



— — r we see at a glance gives the values of u x correspond- 

 ed * )* 



ing to the first particular integral ; and since the two first 

 terms of the function multiplied by C are 1 + 2*, it follows 

 that this function is the generatrix of the second particular 

 integral — in other words, that 



sin-WCT =1+ ^ + ^ 2+ ^^^ ^ 

 (l_f)*(l + f)* 1.3 1.3 ^13j 



* Communicated by the Author. 



