124 Equations of Equilibrium of Elastic Plate under Pressure. 



has the same general characteristics as the curve represented 

 by (89) ; for equation (88) shows that the curve deter- 

 mined by that equation meets the a'-axis perpendicularly at 

 the point where ?/ is zero and x is not zero, and the curve 

 represented by (101) has this same characteristic. 



Now, assuming that y is correctly given by (101), and 

 substituting this value for y on the right-hand side of 

 equation (%^>), we get 



d 2 \, /, 7 \~ 6/7 



Solving this and adjusting the constants so that y and 

 *r- have the same values when ^ = as are given by (89), 

 we find 



*=\-\(}-vT ~ h "- • • • (102) 



The expansion of the right-hand side of this last equation 

 gives 



1 2 1 3 13 17i 39 6 1326 t 



y =x- r *- r - 141* " 288* ^ 864* - 36288* -' 



.... (103) 



an equation which coincides almost exactly with (89). For 



values of a less than unity } the difference between the two 



values of y is certainly very small. Then we may assume 



that the value of y given by (89) is approximately the same 



as that given by (102). 



To find an approximate value of x\ we have therefore to 



solve the equation 



9 9/7 \ 8 / 7 



l- 5x= l(l-l x ) .... (104) 



A close approximation to the root is 



0! = 0-883. 



It follows then that P x is zero or very small when r has 

 the value r , 1 given by the equation 



1 « 2 Eri 2 



51T 2 W = - 883 (1 ° 6) 



Thus we see that the complete membrane determined by 

 our differential equations (in which certain physical difficulties 

 are ignored) has an asymptotic cylinder, the radius of which 

 is given by (105). A section through the middle of the 



