SECTIONAL 'TRANSACTIONS.—A*. 357 
E. —k <u, <k, k> 2, u, diverges to © except when n, lies in a 
certain set of measure zero. This set contains two distinct enumerable 
sets of values of u, for which the sequence is ultimately stationary and 
equal to k in the first case and 1 — k& in the second. It also contains a 
distinct set corresponding to every prime number p for which the sequence 
is ultimately periodic with period p. 
The first four of these headings were worked out by Mr. Chaundy. The 
last one, E, is obtained by investigating the roots of the equations 
wal —RSASR, 
as an equation in u,, and 
‘ Un + p => Un 
as an equation 1n Uy. 
2. With a view to finding expression for u,, as a function of n, we transform 
the difference equation 
u(x + " — u(x) = { u(x) — ki { u(x) +k — r} 
; log 
by putting « = fae a 
thereupon becomes 
(x) Fay) — fly) = f(y). 
This has a formal solution 
Cx je 
@ «| Coe eae an 
where c is an SSieand constant and the coefficients cy are given by reduction 
formule :— 
u(x) =k-+ qyf(y), where g = 2k> 1. The equation 
~+ 
Cntr =n + C26n 2 tn a aicte Cae 
= nm I dee Se I 
era i ae =%) 
It is easily seen ge cn+x is a polynomial ing. We show that S, the 
degree of the polynomial c,, is given by 
S,, = 4n(n +3) — m(n +1) + (n — 2) + 2), 
where m and s are such that 
n=2m7+5, —1<sS2"—y1, 
Using this and the fact that cy =n! when qg = 1, we show that the 
series (2) is convergent for all values of y when g 2 2. ; 
So far the equation (1) can only be completely solved in the two cases 
when g = 2 and q =4. [In the first case f(y) cB t and in the 
second f(y) = cosy =F 
Dr. J. H. C. THompson.—On the dynamics of the crystal lattice, and its 
application to some physical properties of crystals (12.15). 
Born’s method of describing the dynamics of a crystal lattice by means 
of a frequency spectrum is briefly described, and its importance in investi- 
gating physical properties of crystals is discussed. 
The determination of the frequency spectrum for polar crystal lattices 
