of an Elastic Plate under .Normal Pressure. 123 



that T and P are nearly equal. We shall now consider the 



possibility of a zero value of Pi. 



dv 

 Equation (88) shows that ~- decreases as x increases for 



all values of x and y. It follows that y must finally become 

 zero, and it is easv to see that, starting with finite values of 



y and -~, y becomes zero for a finite value of x, and therefore 



y 

 for a finite value of r. Since P x is proportional to - , it 



oc 



follows that Pj vanishes for a finite value of r, and equation 



(82) shows that -7- is infinite when P : is zero. This is, 



of course, a physical impossibility, and, moreover, the whole 



theory fails when -j- is not a small fraction. Nevertheless 



it is interesting to consider this extreme case. 



If P] is zero, the following equation must be satisfied : 



y i i l 9 ■-' 13 3 17 . » , inn x 



• =1 ~ *— 6^-144^- 288* — ••=°- (100 ) 



Now we have already noticed the good agreement between 

 the earlier terms in the series for y and those in the expansion 

 of — (1— #)log(l— %). If we could be sure that all the 

 remaining coefficients of the series in equation (100) were 

 less than the corresponding coefficients in the expansion of 



— or ge( ^ 



as they certainly promise to be, then we could safely assert 

 that the root x 1 of (100) is less than the root of 



-£=^io&(i-<o=o, 



which latter root is unity. 



It is not very difficult to get a closer approximation to the 

 value of #!, as can be seen in the following process. 



/ 7 Y 3/T 

 If the expression x( 1— - x \ be expanded in powers of x, 



it will be seen that the first three powers of x have the same 

 coefficients as the corresponding terms in equation (89). 

 Moreover, the curve represented by 



y=^l-l.vj' (101) 



