THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 145. 
(There are, of course, any number of equally suitable modifications; instead of 
e-“" we might take e~“ or 1/(1+«°), for instance.) A solution of the preliminary 
| problem of normal traction equal to «/(R’+«’)! on the face z= is thus obtained 
in the form 
ae cosh kh ; 
ae =I of { [cama 
sinh kh H Ke-«h 
deck, cosh = =e ea 
# k (sinh 2«h + 2h) ean tT Hed Ke K } 
G h kh + 2«h sinh Kh : 
4u6= | are) HE etn ee 
= a ear (sinh 2«h —2«h) pee oa 
sinh kh + 2«h cosh Kh L Me-«h 
Ea ae, he-4-"e bg y aay 
« (sinh 2«h + 2xh) Ce aaa Ke kK " ( ) 
The solution of the original general problem is found by multiplying by 
F(x, y)/27, integrating with respect to a’, y’ over the area A, and finally taking 
the limit for «=0. But a glance at the forms near «=o of the functions in (14) 
shows that the triple integrals are absolutely convergent, it being supposed that. 
—h<z<h. Hence we may integrate with respect to x’, y’ first, and by a well- 
known theorem the limits for e=0 may then be found by simply putting «=0 in 
the integrands, provided the resulting integrals are convergent, as they manifestly 
are. 
This gives the value of ¢, for example, in the form 
Ja | [Fle area, 
but, always provided—h<z<h, we may change this if we please into 
[[Fesyacay |) Boode. 
Finally, we may with great advantage confine our study in the first place to 
what is usually spoken of as a wnt element of normal traction at (a’, y’, h). The 
area A enclosing this point is diminished, and the intensity of traction increased 
without limit, so that | | F(a’, y')da'dy’ remains equal to unity. The resulting 
solution is simply that of (14), but with e put equal to zero within the integral 
signs. 
As we have just seen, the solution for the general case can at any time be 
found from this elementary solution (15) by multiplying by f(a’, y’)/27 and integrating 
over the area A. 
4, Flexural and extensional components of the strain. Disadvantages of the 
solution in definite integrals. 
In the elementary solution each of the potentials ¢, 6 may with advantage be 
decomposed into an odd and an even part in z ‘Thus, for an element of normal 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 24 
