178 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
The source itself is similarly given by the temporary values 
rp. al 
Vie 2%? 
, a BN K(Z- 
G = - Sp) Fee OTR 
P I 
eee Dr 
Thus, when the source is included, the provisional values of \’, 6’, ¢’ are as in (80), but 
with the first lines altered, 
in wy’ to +° +1 inh K(z— 2’) 
2%? 
” - » + = sinh K(2 es z) Jk o G 5 (81) 
K 
» oO, +—sinhk(z-2)F = cosh K(z —z’) 
2? K 
25. Solution of the problem of internal force parallel to the faces. 
From these expressions the solution in the form of definite integrals, and finally of - 
series, is obtained as in the previous cases. After the explanations already given, it 
will be sufficient to write down the final results. For the transitory part of the 
solution, 
y= (a+ US sinh xz’ sinh xzGo«R, (x a pos. imag. root of cosh xh) 
kK K/L A 
= 6 + De cosh xz’ cosh xzGyxR, (x a pos. imag. root of sinh xh). 
K KIL 
= Sinh c@GokR { — «2 cosh x2’ + 5;(cosh 2xh — a) sinh xz 
= 2S same as previous line multiplied by ( — cosh 2«h) 
where « is a zero of sinh 2«h — 2xh, with pos. imag. part. 
With 
Lhe cosh xzGyxR { pea Il a 
= eo) Ae  EAC h Ne h 
ce) : ( ) Gao kz sinh Kz + 5 (cosh 2«h + a) cosh xz | 
= Dat same as previous line multiplied by cosh 2«h | 
where « is a zero of sinh 2xh + 2xh, with pos. imag. part 
the functions of (80), altered as in (81), we omit the factor J,«R, and then find its 
expansion near «=0 to contain terms of negative degree in x, say A/x®+B/c. The 
