1887.] deformation of elastic plates. 149 
Introduce now three functions depending on the stretching of 
the middle surface, viz. the extensions of the line-elements 1, 2 
and the sine of the angle which 2 makes with the axis y after 
strain. Call these o,, o,, a, then 
0 
+a)=%, md+e)=2, ute) =) 
(n, + 2,7) (1+o,) = | 
Also introduce six functions depending on the bending of the 
middle-surface, x,, \,, T,, Ky» ,, T, defined by the equations 
sins ea tie on, 
i 5 Ae 2 Og ee (5), 
where X,, 7, are got from these by cyclical interchange of the 
suffixes 1, 2, 3 and «,, X,, T,, are the corresponding quantities with 
8 in place of a. 
In case the middle-surface be unextended, X,, «, are the curva- 
tures of lines through P initially parallel to a, 8, and «, depends 
on the angle one of the directions of principal curvature of the 
middle-surface after strain makes with the line initially parallel 
to a, so that «,r, + «,’ =the measure of curvature. 
Now multiply the set of equations of which (8) is the type by 
I, ™,, 7,3 1,, m,,,; 1,,m,,,; and add each time and perform 
similar operations on the second set, we obtain six equations, viz.: 
@ 
maT +)+%@+u) +o, 
ov 
ap (2 +) +7, (@ +X) 
0. 
ap (wt) +e, (y+), 
AAP gi ak Sey catalyse Spann areas (6). 
Jy TY TDtA E+ Mtoe 
0 
py aE Tw) tT (@ +) +0, 
and 
Ow 
Gy e+e) HG (y +2) 
