19 



In the ordinary notation of elliptic integrals we have 



I 



— = F(e,<t>) 



2a 



f 



where sin 6 = — and cos <f> = 



2a h + f 



or, more neatly, in the inverse notation, 



/ 



— en (u, k) 



h + f 

 I e 



where u — — and k = — 

 2a 2a 



I 

 According to Euler's theory, the maximum value of — 



2a 

 when the load is axial is \ ir. For eccentric loading, there- 

 fore, the value must certainly be less than this. In cases of 



e 

 any importance to engineers the value of — is quite small, 



2a 

 commonly of the order of '1 or less. 



Now the function c n (u, k) may be expanded in series, 

 and we have (Cayley, Elliptic Functions, p. 57) : 



cn(u,k) = 1 -t+ (i + 4ft»>p-.(I + 44* i + I6ft*)|j+ 



But we know that 



?t 2 u* u G 

 coe«=l- j^+ jpjj+— - 



I e 



And as — is never much greater than 1 and k or — is quite 

 2a 2 a 



small, it follows that it is quite sufficiently accurate to write 

 cos u in place of c n (u, k) and thus obtain the result 



/ i 



= cos 



h + f 2a 



from which h may be simply determined. In applying this 



I 

 formula, of course — represents the circular measure of the 

 2a 



angle, the cosine of which is to be taken. 



