TBANSACTIONS OF SECTION A. 



699 



be exerted on the oscillator to keep the vibration simply periodic, and if the im- 

 pressed force be of a proper period any period of vibration is possible. To solve 

 the problem a system of equal incoming waves is superposed on the outgoing ones, 

 and then the simply periodic vibration is possible without any impressed force, and 

 this condition then gives the free period of vibration. 



7. Magnetic Action on Light. By J. Laemok, F.B.S. 



This Paper was ordered by the General Committee to be printed in extenso. 



See Reports, p. 335. 



On a Special Glass of Generating Functions in the Theory of Numbers. 

 By Major P. A. MacMahon, B.A., F.B.S. 



9. On Agreeable Nimibers. 

 By Lieut.-Col. Allan Cunningham, B.E., Fellow of King's College, London. 



A number, N, of which the ?« digits on the right hand are the same as the vi 

 digits on the right hand of its nth power (N°), when both are expressed in the 

 scale whose radix is r, is styled an Ageeeable Ntjmbek of the m.th order and nth 

 degree in the i-ary scale. When the agreement of N, a number of m digits, with 

 its nth power, extends throughout its m digits, the number N is styled a Complete 

 Agreeable Number. The analytical condition is 



N" - N must be divisible by ?'™. 



The properties of these numbers are investigated in a quite general manner 

 applicable to any scale of radix ;• ; and simple rules for their computation given. 

 These rules are completely reduced for the denary scale to their simplest form, 

 and the auxiliary quantities are tabulated. Computations of complete agreeable 

 numbers are given in detail for the denary scale. Tables are given of all agreeable 

 numbers to the fifth order, and in some cases to the tenth order. 



Example. — The numbers, N, of ten digits (shown in table below), and also 

 the numbers of fewer digits obtainable therefrom by erasing one or more of the 

 extreme left-hand digits of N, are complete agreeahles in all the degrees n stated 

 in column n; and are, moreover, the only complete agreeahles (often digits, or less, 

 ending in 1, 2, 4, 5, 6, 8) in all those degrees, except when n has the critical forms 

 named in column n', in which case there are a number of complete agreeahles 

 (increasing rapidly with the value of n' and with the number of digits of N). 



MONDAY, SEPTE3IBER 18. 

 The following Reports and Papers were read : — 

 1. Beport of the Oommittee on Earth Tremors. — See Reports, p. 287. 



