integral by Biot-Savart. Both ways are found in literature, the first one in papers by 

 Kawada [1] and Lerbs [6], the second one in papers by Maikapar [2], Schubert [3], 

 Guilloton [4], Strscheletzky [5] and Lerbs [6]. Of main interest are the axial and 

 tangential components which are induced on a propeller blade. If this blade is replaced 

 by a lifting line for the present, the components of the velocity induced on the radius 

 x — r/R of this line are represented by the following expressions : 



/ w a \ dG dxo 



dl — J = Yda " (1) 



\ v / dxo x — Xq 



and correspondingly for the tangential component if the subscript a is replaced by t. 

 This element of velocity is generated on the point of reference x by a semi-infinite 

 spiral vortex line of radius x — r /R and of circulation dG/dx . By G = T/ttDv 

 the non-dimensional bound circulation on the radius x is denoted. The "induction 

 factors" i represent the ratio of the velocities induced from a spiral vortex line to that 

 of a straight vortex line. They are introduced to make possible a finite representation 

 of the element of the induced velocity if x —> x . 



On the basis of the integral by Biot-Savart the induction factors may be ascer- 

 tained in the following way [6]. As a first step, the expressions for dw a and dw t as 

 established from Biot-Savart's law may be considerably simplified since both of these 

 expressions may be reduced to a derivative of an integral over 1 /T, T being the distance 

 between an element of the spiral vortex line and the point of reference. One finds that 



l dr ^, d f x / l 

 dw a = / . r — / [ — )da (2) 



dw t 



(3) 



where k ■=. r tan /?,-, /?j being the pitch angle of the vortex line. The integral may 

 be solved if \/T is expressed first by an integral over the Bessel function J applying 

 the Lipschitz integral and if 7 is then developed into a series of Bessel functions of the 

 first kind by means of Neumann's addition theorem. In this way one is finally led to 

 the Hankel integral which makes possible the following representation of the induction 

 factors : 



Exterior field (x > x ) 



ia = 2[--l)(-^—)A (4) 



\xo I \ tan (3 { ) 



Xo\ / z \ 



- )z 1 + 2 A) (5) 



x ) \ tan fa ) 



mz 

 A = J ' ml'rm [ ]K m A -J (6) 



tan j3i 

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