of an Elastic Plate under Normal Pressure, 121 



Next, putting 



5 X 4 =54 



(88) 



128A 8 B" v. (85) 



and /i 2 fi = £ 



we find that 



dtf~~W ( } 



The complete solution of this equation contains two 

 arbitrary constants, and we can see immediately how one of 

 these constants is involved. For, putting 



6 = Cfy, s 1 = CKv, . . . . • (87) 

 we find that [86) becomes 



dfy _ x 



dx 2 ~ y 



an equation which does not contain C. It follows that C 

 is one of the arbitrary constants. 



Now there is only one boundary condition in our present 

 problem. It might be, for example, that F 1 has a given 

 value when r= a, the radius of the boundary circle. The 

 one constant C which occurs in the relations between x, y, s u 

 fi, is quite enough to enable us to satisfy this one condition. 

 Then we need only a particular integral of equation (88) of 

 the right type to solve our present problem. 



Equation (88) has a solution expressing y in an infinite 

 series of integral powers of x starting with the first power. 

 This solution is 



1 t 1 , 13 17 , 37 fi 1205 7 



It is interesting to notice that, as long as x is a fairly 

 small fraction, the series for y differs very little from 



1 2 1 3 1 4 1 5 



2 6 12 20 ■••' 

 that is, from — (1—x) \og e (1— #). 



The radial tension is 



r dr ~ r 2 



-M^i-'i'-ig^-l' • (90) 



P being, of course, the value of Pi at the centre of the 



