THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 161 
Now since | 
(wen!) | [GocRMe’, yydaldy/ = - 2atle, 9), 
] . 1 °, Ie , t , 
one value of = eae “Ff is = a] [Goxkre y da‘ dy’ . 
“ye . I¢7 . 1 + - / € 
Similarly one value of vy is 3, | [tvs (R/2h) f(a’, y')da dy ; pie (38) 
1 loo), BR - 1 br teas, 
Hence for the first part of 4ud we obtain 
moe y )da'dy’ + alae + mall? (RY fA(x’, y')dar'dy' | 
cosh xh hee Kz On 
qe (- ) ch(cosh 2kh — pf. | Be ou aay 
(39) 
which agrees with our previous solution, the element of which is given in (20). 
Further, the results obtained at the end of § 14 clearly agree with what we should 
get by expanding the function of « in (37) in ascending power series and interpreting. 
As an example of this use of equations (37) we may find to a first approximation 
the value of 2; at points not very close to the edge of A. 
From (5) 
Be) = - (4a) - 5 £ (4u8) + 28 (420) 
— (2eosh ch + 2«h sinh «h) sinh «Kz — 2«z cosh Kh cosh Kz - 
Pi sinh 2«h — 2xh 3 
4 (2 sinh kh + 2«h cosh kh) cosh xz — 2x2 sinh kh sinh xz _ / 
~ sinh 2xh + 2«h 
= (8h? —2 )2f/2h +f. 
Thus 
gz = { (8h? —2)z/4h? + 1/2} f(x, y), ‘ : : . (40) 
and we verify at a glance that this gives the proper values at the faces. 
16. The problem of tangential face traction. Solution for an element of traction. 
We will now pass to the problem in which the given surface traction is tangential. 
Taking the direction of the traction parallel to the axis of x, we may take for 
conditions 
wm =f(e,y) on z= +h 
m= O on 2=— i, > . 5 : - (41) 
y=u= 0 on esi | 
According to the method explained in § 3, we begin with the function «J,(«R) in place 
of f(x,y), and determine potentials , 6, @ giving 
ay d26 Pp Bp l 
dydz * dudz a ogee =a ee on z=h 
= 0) on z=-—h 
CUI Rac cag dp _ : - (42) 
= 5 EP = 
daz dz i dydat dy dz a og dedy 0 on 2 +h 
a6 ah hp Bp _ 
dz aes OTs 
TRANS. ROY. Soc. EDIN., VOL. XLI. PART I. (NO. 8). af 
= 0) on z=+h 
