Elastic Eiiuilibrium <ii Semi-li. finite Soli.l. 



29 



In the present case, we get , 



^ [v) = + I. — ^ ^ . 



~ 2a7- 



It may easily be shown that the vahie of }S''{v) is a pure imaginary 

 quantity for tlie vahies of v given in (G7), with positive sign for (i) 

 and witli negative sign for (ii). Therefore we liave to take 



r{v) = + I -^- '- 



'2ar 



for 

 and 



for 



in which 



7}i = —)l , 



r(y) = -t, — 



j 2ar 



m = — (m' + 1), 

 0-<^<l. 



Hereafter we shall, for the sake of simplicity, take into 

 account only the value of v which will be ol)tained by putting 

 71 = 0, n =0 in (i) of (07) \va. 



c = (t), -f- d(o.^. 



By this convention tiie term mni disappears and the vahie of ^'C^) 

 is to l)e taken as 



(68) 



iP'(v) = +i 



"Aar 



Thu^ 



P^i-rco^ ^ = -^<o, _ JIz:«^ . _A_{rr,-aj,:(v)]. (69) 

 / BV^M^ 2a 2aV riv) I '' "^-7 ^ ' 



Similarly we have 

 sin-^ 





if VÜ) y 



