188 
MR. L. N. G. FILON OX THE ELASTIC EQUILIBRIUM OF 
The fifth order terms give 
which imply tlie four relations 
(X + 2p) - pE — 0 . 
O 
2 (X -j- jU,) F -j- SpE'^ -f- (X -j“ 2jU,) 4'?^^ — 0 
4E (X + 2/x) + SjxY + 3E' (X + p) = 0 
4E (X + /x) + 3E' (X + 2 /x) = 0 . . 
(83), 
(M). 
(85). 
There is, however, a further relation to he satisfied among these constants, and 
that is obtained as follows. If we proceed to write down the expressions for rr 
and rz and to put in them o' = a, we sliall obtain expressions of the form 
rr = algebraic polynomial in 2 + series of cosines of mTz[c, 
rz = algebraic polynomial in ,2 + series of sines of mrzlc, 
where the coefficients of cos mrzjc, sin mrzjc, contain the two undetermined con¬ 
stants and An. 
We may now proceed to expand the two polynomials in series of cosines or sines 
of mrzfc. Equating then the coefficient of each cosine and sine to zero, we can 
make rr and rz zero over the whole of the curved surface, and at the same time we 
obtain two equations for A^ and Ao. 
But it is clear that, if the Fourier expressions in the second case are to be con¬ 
tinuous, then the algebraic polynomial part of rz must reduce to zero when 2 = rb c, 
otlierwise its expansion in sines of mrzlc is discontinuous, and at the perimeter 
of the flat ends the shear is discontinuous. Tins introduces infinite stresses at this 
point which render the solution inconvenient. 
Now we have at our disposal nine constants; these have already been made to 
satisfy the seven homogeneous equations (78), (80)-(85), and therefore we are free 
to make them satisfy an eighth homogeneous equation. 
Choose then the constants so as to make (polynomial part of [(/a/cA -fi dayV/?’] when 
r — a,z — ± c) zero, and we have 
Dac -f Ym?c + Fac® + DAc + E'ac^ = 0 
( 86 ). 
If now we express all the other constants in terms of the constant E, we find : 
