20 



I 

 If — =1 and k — 01, the error made by computing h 

 2a 



from this simple formula is only 0'2 per cent. If k = 0'2 the 

 error amounts to 0*8 per cent. If k = 0*3 the error is 1*8 per 



I 

 cent. If — < 1 the error will be still less. 

 2a 



The error made in using this simple formula does not. 



I 

 become appreciable unless — is considerably greater than 



2a 



1 and k has also a considerable value. In such cases we may 

 write the equation to be solved in the form 



/ 



k en (u, k) = — 



2a 



( l \ f ( 6 \ • 



where u = j — I and — are known and k I = — is to be 



\2aJ 2a \ 2a J 



determined. A numerical solution of this equation may then 

 be found by the aid of the published tables of the values of 

 elliptic functions.* But for the practical problems of engineer- 

 ing this process is not required, and the simplified approximate 

 formula may be adopted. 



An inspection of a table of Elliptic Integrals shows that 



I 

 the integral representing the value of — is < 1 for all Values 



2a 



of k provided that the value of <\> is not more than 49°, 



.. ... „ .. / 



i.e., if is not < cos 49° 



h + f 



or provided that h is not more than '524 /. 



Numerical Illustration. 



Suppose that we wish to compute the deflection of a steel 

 column in the form of a British standard steel joist 8" x 4" 

 and 9' 6" long, under a load of 30,000 lbs., the line of action 

 of the load being 0*5" from the axis of the column. 



From the tables of standard sections we find the least 

 value of / is 3*57. 



