344 SECTIONAL TRANSACTIONS.—A*. 
and generalise for the many-rowed partition into the form 
ae ehes | a e. Ape tee | : 
Ey 
v G51 g, 6341 
| 
4 2 € 
MGe=5 PCa a ¢ 
This formula extends to various other cases of specially-conditioned partitions and 
includes MacMahon’s formula (1) as a special case. 
Mr. E. M. Wricur.—Asymptotic Partition Formule. 
Hardy and Ramanujan found an asymptotic formula for p(n), the number of 
linear partitions of n, by studying the properties of the generating function 
co fos] 
fila) = T(t — 21 = 14 S pine, 
l=1 n— 1 
and so finding the value of ‘2 
| fule)de 
ntl 4 
JC 
where Cis the circle | | =1 — = 
This method can be extended to determine an asymptotic expansion of p,(n), 
the number of partitions of n into k-th powers. The generating function is 
fidx) a ic ai = 1+ Spilnyer. 
i 
The transformation theory of f,(z) has to be developed by lengthy analysis. We 
then obtain expansions of p,(m) in a series of integra] functions of »'*, and in a 
series of descending powers of n/(k+), We can also find an expansion of g(n), the 
number of plane partitions of n, in descending powers of n3, by means of the generating 
function found by MacMahon and Chaundy. 
It is interesting to consider ‘ weighted partitions ’ of n into the parts y;, Us, . 
each weighted 2, Ao, . . . . , So that the generating function is 
co —1 co 
Il ( fess nent) =1+4+ > pan). 
= 
n=) 
s 9 
The most interesting case appears to be that in which p(l) =1, 4;=a>0, and the 
generating function is 
foo} a co 
| bey i ep Pee ate i : 
1 ( a) > Paln)e" 
n=] 
lf a <1, the asymptotic expansion of p,(7) is found as in the problems already con- 
sidered. If a>1, new methods are applicable. a, is the root of a certain transcendental 
equation ; if l<a<a, p,(n) has an asymptotic expansion, which may be found by 
an application of Cauchy’s theorem. If a>a, the asymptotic expansion becomes 
convergent, and 1s equal to p,(”) for all positive integral values of n. In the latter 
case, the expansion may be found by ‘ elementary ’ methods. 
Prof. A. J. McConneLi.—Applications of Differential Geometry to Dynamics. 
The configurations of a dynamical system of N degrees of freedom can be repre- 
sented geometrically by points in a space of N dimensions, and consequently the 
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