Of the four approximate equations tried, equation (a) shows the best agree- 

 ment with the exact equation (6). The agreement is excellent for all values of 

 1/d less than 2. The mean of the absolute deviations throughout the range of 1/d = 

 0.1 to 2 is 0.5 per cent. The maximum discrepancy is less than 3 per cent for 

 1/d = 2 and is considerably below 2 per cent for all values of 1/d equal to or less 

 than 1 . 6 and is less than 1 per cent for all values of 1/d less than 1 . In fact, 

 Eq. (a) gives better agreement with Eq. (6) than does Eq. (7) from which it was 

 derived.' 



Equation (a) can be put in a more convenient form by substituting values of 

 y, x, and oc from Eq. (6 1 ) and (9), whence 



P = 



2 - 424E , (h/a)* 

 (1 - (T 2 )* 



-^-- 0.447 V"hA 



or (o>= 0.3) P = I' 60 E < t/d) ^ 



J - 0.45 Vt/d 



(a«) 



Again it might be pointed out that the usefulness of Eq. (a') is not con- 

 fined to the neighborhood of the origin. It is a simple equation, independent of 

 n, which determines p with a high degree of accuracy for a large range of values 

 of 1/d and t/d. Equation (a 1 ) may replace equation (6) in all practical computations 

 where 1/d is not greater than 2. When 1/d is less than 1 the discrepancy will be 

 less than 1 per cent .J 



In order to facilitate the practical application of the results derived 

 above, one may use to advantage a graph, Fig. 4, that contains the polygons (they 

 appear as curves) for the values OL = 2, "4, 6, to 20, within the 



— 6 



range up to x = 6 x 10 corresponding to the ratio a/1 from approximately 2/3 to 

 6 (1/d = 0.8 to 0.08) and to the ratio h/a up to about 0.0045. The straight lines 

 are computed throughout by the complete equation (6). In the neighborhood of the 

 origin, the polygons are made up of curves determined by Eq. (12). The corre- 

 sponding angles of the polygons are connected by dotted lines, forming quadrangular 

 areas for the various wave numbers. The separate vertical straight lines correspond 

 to the constant values of 1000 h/a written beneath; the values of the ratios 

 "tfa/1 are written on the sides of the polygons. On the axis of ordinates, a second 

 scale is represented beside the y-scale which, under the assumption of an elastic 

 modulus E = 2,125,000 kg/cm 2 (30.22 x 10 lb. per sq. in.), gives the value of 



k = p a/2h = yz-p y = 2 -34 x ^ T < 1 3) 



up to k = 1900 kg/cm 2 (27,000 lb. per sq. in.) The values in the range of or- 

 dinates above this point correspond to the observations of Tetmajer on non-elastic 



