THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 139 
practically insensible when the distance from the source is a moderate multiple of the 
thickness of the plate. In the following pages we shall deduce analogous solutions for 
‘sources of strain’ of the different types which may exist in an elastic solid, and develop 
these solutions in various directions. The corresponding development of the present 
solution is extremely easy, but would carry us too far. We merely mention that the 
‘main current’ in the hydrodynamical problems corresponds to the ‘ principal modes 
of strain,’ the determination of which is the object of the theory of thin plates. But 
there is one important. distinction in the two cases. In the flow problems the 
exact conditions defining the ‘main current’ can always be found, and are indeed 
obvious; on the other hand, the analogous conditions in the strain problems can only 
be found by approximation. 
(k) The following conventions seem to be very generally adopted, but to prevent 
any risk of ambiguity they may be stated explicitly here. Consider any continuous 
plane area A bounded externally by a closed curve C,, and internally by one or more 
closed curves C,, C,, etc. At any point EK of a bounding curve let Hx, Hy be drawn in 
the directions of the rectangular axes of coordinates. Let Ex, Ky be turned through an 
angle «, which will be taken as positive when the rotation is counter-clockwise, until 
they coincide with Hé, Hy, the direction of Hé being that of the normal at E when 
drawn from within A towards the boundary. Hé, Hy will be taken to be the positive 
directions of the normal and tangent at H, and if f(a, y) be any function given within 
A, 2 and a will be used to denote the rates of variation of f per unit length in these 
positive directions. 
The curvature at E is = and is denoted by 1/p. p is therefore positive when, in 
order to reach the centre of curvature from H, we have to proceed into the area A. 
If we suppose the figure traced on level ground, a person proceeding along the 
boundary in the positive direction will have the area on his left, and the curvature 
will be positive when he is rotating about the vertical in the counter-clockwise sense. 
The following formulz relating to differentiation along the are and normal will be 
much used in the later sections of the paper. Suppose the axes of x and y to coincide 
with the positive normal and tangent at a point O of the bounding curve. At a 
neighbouring point E (x, 7) on this curve 
ne coun tf acting 
dn dx dy : 7 : . . (i) 
ae — sin Ds cose | 
ds dx dy 
By putting x, y, « equal to zero, we have at O 
Cor hp he he 
Gaus ay , 5 z F ; (ii) 
