<DOC>
<DOCNO>
1047
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<TITLE>
the bending strength of pressurized cylinders .
</TITLE>
<AUTHOR>
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<BIBLIO>
</BIBLIO>
<TEXT>
be described here is attributed to the russian
investigator v. g. galerkin, whose original papers are
inaccessible to the present writer .  his knowledge
of the method is derived from a description given in
a paper by e. p. grossman .  grossman states
that the method was given by galerkin in his treatise
p. 897), and that applications to oscillation problems
were first made by v. p. lyskov .  it is pointed
out by grossman that galerkin's process in applications
to mechanics leads to the same results as
lagrange's principle of virtual work, but employs a
special co-ordinate system .
  the method of galerkin belongs to the same general
class as those of rayleigh and ritz, for it seeks
to obtain an approximate solution of a differential
equation with given boundary conditions by taking
a function which satisfies these conditions exactly,
and proceeds to specialise the function in such a
manner as to secure approximate satisfaction of the
differential equation .  the selected function is
a linear combination of n independent functions, and
the coefficients are determined by a process of
integration .
  the galerkin process can be considered from two
points of view, (a) simply as a means for the
approximate solution of differential equations, and
treatment of problems concerning the statics and
dynamics of elastic and other deformable bodies .
these two aspects are treated separately in parts 1
and 2 of the paper respectively, and will now
be briefly discussed .
which satisfies the boundary conditions, in
the differential equation be .  since the result should be
zero, is the error in the differential equation .
then the galerkin process consists in choosing the n
coefficients in the function in such a manner that
n distinct weighted means of the error, taken throughout
a certain range of representation, shall all
be zero .
as a generalised force, and the multipliers used
to weight the errors are the virtual displacements
corresponding to increments of each of the generalised
co-ordinates in turn .  thus the vanishing of the
weighted mean is here interpreted as the vanishing
of the virtual work in the appropriate displacement .
  the degree of accuracy attaindd can be increased
indefinitely by increasing the number of independent
functions employed, but this entails a great
increase of labour .  however, when the functions
are well chosen, an excellent approximation can be
obtained by the use of a very small number, as is
sufficiently shown by the examples included in this paper .
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