Solving Equation [77a] as a quadratic in z and expanding the radical, 



1 1 / 4c2 

 - + - 1 - 



2 2 \ ,.2 



'/2-, 



1 1 



z z' 



1 - 



by the series, (1 -a?) = 1 + a; + ; hence 





1 Ae"? 



^2 



; 1- 



1 + — + .... = 1 + — + . . - 



„2 „f2 



Hence, as far as terms of order I/2' , 



dw ., iV 1 



— , =v e-'y + 



dz 2n- s' 



^■^ 



277 



Ae'''? + fy (c2 e-'y - a2 e'-y) 



Thus the constant 6, in Equation [74j] with 2 replaced by z'here has the value 



iV 



b^=--- he'-^ -V (c2 e-'y - a^ e'^), 



277 



[77f] 



and, upon substituting in Equation [74ni] and selecting the real part as indicated by the 

 symbol (/?), the torque per unit length on the cylinder about an axis passing through the 

 origin of coordinates is found to be 



W = - 277 pc^ {;2 sin 2y + p Art/ cos (77 - y). [77g] 



(See Reference 2, Section 7.50, where the sign of y is reversed; Reference 4.) 



176 



