We find the solution 
SECTIONAL TRANSACTIONS.—A*. 343 
functions and that of lacunary trigonometric series. For example, we have the 
following theorem : 
If f(é) is of class L*, where 1<k<co, if {Aj>=0, Ai, Xo, . .} is any Sequence of 
positive integers for which A,,,1/A,,>q> 1, and if S, (t) is the 8th Cesiro mean of 
rank p of the 9-series associated with f(t), then the upper bound of | 8, %¢) |, for 
variable m, is also of class L*, and the ratio of the integral of the kth power of this 
function to the integral of the th power of | f(t) | between the limits 0 and 1 lies 
below a number dependent only on k, qg, and 8. 
With 5=0 this theorem describes properties of a function derived from the 
partial sums of the -series. The function 
© 2) ; 
o: Sa, ,(4)—Sn,, (0) | 
m=0 A 
where §,(¢) is written for S,,(¢), formed from the partial sums in an entirely different 
manner, has the same properties. 
Mr. J. M. Wurrraxer.—The Composition of Linear Differential Systems. 
The paper is concerned with the problem of resolving a linear differential system, 
consisting of a differential equation together with boundary conditions, into two or 
more systems of lower order. The product of two such systems is defined, and it is 
shown that the Green’s function of the product system.is the product (by composition 
of the second kind) of the Green's functions of the component systems. It follows 
that the two products of a system with the system adjoint to it are self-adjoint, and 
it is shown that the solution of the original system can be found in terms of the eigen- 
functions of these two product systems. 
Mr. T. W. Cxaunpy.—Partition-generating Functions. 
MacMahon, by imposing on a partition the condition of arrangement in monotonic 
order, was able to define many-rowed partitions (or ‘ plane’ partitions) which were 
not mere obvious combinations of ordinary (7.e. one-rowed) partitions. 
Thus 4), DOV AL 
D2 | 
1 
in which both rows and columns run in non-ascending order, is a 3-rowed partition 
of 15. MacMahon, by his very difficult algebra of ‘lattice functions’, evaluated 
generating functions for plane partitions of various types, culminating in the remarkable 
formula : 
co 
II (1—2")-* (1) 
r=1 
for the unrestricted plane partition. 
I base my treatment of these partitions on certain recurrence-formule in which 
the monotonic principle is inherent. Thus the generating function ¢, of ordinary 
partitions, in which the first part does not exceed a satisfies the recurrence-formula 
b= & , 
which leads at once to. Euler’s formula for ¢,. Similarly ¢,, the generating function 
of two-rowed partitions in which the first parts of rows 1, 2 do not exceed a, 6 satisfies 
the recurrence-formula ; 
é.=> >” as 8 . 
8=0 r=s 
E,=€, ‘ 0&3 . Se ? 
