276 Prof. GayUy, Table of A'-'O" 4- n {m) up tom = n^ 20. [Oct. 27, 



The following communications were made to the Society: — 

 (1) Pkofessor Cayley, Table of A^'O'^-^H («i) up to m=n=20. 



(Abstract.) 

 The differences of the powers of zero, A'"0" present themselves 

 in the Calculus of Finite Differences, and especially in the appli- 

 cations of Herschel's theorem, /(e*) =/(l + A) e*", for the expansion 

 of a function of an exponential. A small table up to A^^O^", is 

 given in Herschel's Examples (Cambridge, 1820), and is reproduced 

 in the treatise on Finite Differences (1843) in the Encyclopoidia 

 Metropolitana. But as is known, the successive differences AO", 

 A'^0", A^O", ... are divisible by 1, 1.2, 1.2.3, ... and generally 

 A"'0" is divisible by 1.2.o...m, = 11 (m) ; these quotients are 

 much smaller numbers, and it is therefore desirable to tabulate 

 them rather than the undivided differences. A table of the quo- 

 tients A"0"-f-n (m) up to w = 7i = 12 is given by Grunert, Crelle, 

 t. XXV. (1843), p. 279, but without any explanation in the heading 

 of the meaning of the tabulated numbers G^ [= A"0* -^ 11 (w)}, and 

 without using for their determination the convenient formula 

 C,^"^^ = n.C„* 4- O^.j given by Bjoiiing in a paper, Crelle, t. xxviii. 

 (1844), p. 284. The formula in question, say 



= m 



n(m) n(m) n(m-l)' 



is given in the second edition (by Moulton) of Boole's Calculus 

 of Finite Differences (London, 1872), p. 28, under the form 



A^'O'^ = TO (A'''''0""' -1- A^'O""'). 



In this paper the author extends the table of the quotients 

 A"'0"'-;-n(m) up to m = w = 20, the calculation having been effected 

 by means of the foregoing theorem. 



The paper will be published in extenso in the Transactions of 

 the Society. 



(2) Mr W. M. Hicks, M. A., On the problem of two pulsating 

 spheres in a fluid. 



1. By pulsation is meant a periodic change of volume in the 

 same manner as by vibration is generally understood periodic 

 change of position. The name was given by Bjerknes, who has 

 considered the problem of the motion of two spheres in a fluid 

 which change their volume, in a series of papers read before the 

 Scientific Society of Christiania in 1863, 1871, and 1875*. He 

 approximates to the apparent forces acting on the variable effective 



* See also an abstract of the last in the " Repertorium der reinen und ange- 

 ■wandten Mathematik, " BJ. i. ' 



