THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 185 
Il. Flexural terms. 
The flexural solutions are of the form given in (23), or in polar coordinates 
d 
~ dp nee ee ato) 9/1-3 2 on | 
id =(a+1]l)@k— fy °F) P2ge hz) y2F | | (94) 
ae ae ‘x 
w= (a+ 1)(F—32 7 2F) + 2(22 -h?) 7 °F 
(i) F=x(p) = 4p? log — 3p” 
(i1) 4 p log a COS w 
(iii) F=3(p—2p log i) sin o 
: 1 ; 
. F= —m+2 . 
(vy) 4m(m — ine ee eel 
1 Aa | : 
1h —m+2 
(v) me Dp? sin mo | 
i) F=lo¢e Ape 
Cay ale 
(vii) F=p-” cos mw 
She : m>0. 
(vill) F=p~” sin mH 
For the force with components P,, 2, , Z, the coefficients of the above solutions are 
the following, in each case divided by #27uh?. 
(i) Z 
(il) —2, cos w,P, +2, sin o,Q,+Z,p, cos wo, = — X42, + Z,x, 
(ili) — 2, sin w,P, —2z, cos o,0,+Z,p, sin o,= — Y,2z,+ Z,y, 
c A m—1 os m—1 at m Aas 
(iv) —2 mp," cos mw,P, +2,mp,"" sin Mo,Q, +p,” cos muw,Z, 
v) Same as (iv) with cos mw, changed to sin mw, , and sin mw, to —cos mw 
1 8 1? 1 1 
» 
(vi) -dap,P, + | Jp2 +3? - }e2+ (22-1) |Z, 
a+1 y 
(vil) [ sam oes 1 ee | 42,? — th2, is = (44° — h?2z;) } a” | cos mw,P 
as M+) _ iW iii sin mw,Q 
[ = a 1 fr | ‘abe 
1 m+2 yy 2 2 ys 1 m 
es [ are +7 4 \ th? — 42,7 + 1Ge — h?) ben, | cos mw,Z, 
(vili) Same as (vii) with cos mw, changed to sin mw, and sin mw, to — cos mo. 
30. Types of deformation conveying a given resultant stress. 
In these formulze we remark at once a striking relation between the forms of the 
displacements u,v, w in the various solutions, and the multipliers of P,, 2,, Z, in 
the coefficients of the solutions. 
In I. (aii), e.g., these multipliers are p,”-*cos mo, ,— p,""'sin mo, ,0, which are 
simply the displacements of I. (v) with sien of m changed, and consequently suitable 
for space containing the origin. 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 29 
