CONFOEMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 451 



It must be remembered, however, that this transformation has been constructed 

 solely with a view to the angular configuration, and that it affords no guarantee that 

 the stream-lines corresponding respectively to w > c and w < c will not be merely 

 parallel instead of being in the same straight line. 



There is, in fact, the possibility of a configuration of the character indicated in 

 fig. 4, and it must now be shown that this possibility can be provided against by a 

 suitable adjustment of the parameter k. 



16. Significance of the Early Terms in the Expansion of w for Great Values 

 of z. If in a transformation such as that of formula (29) the variable w be supposed 

 to have its modulus large, and an expansion be made in ascending negative powers 

 of w, the result is a formula of the type 



dz = V- 1 {l + Sw- 1 + VDw- s }dw > ' . . (30) 



where negative powers of w beyond the second are neglected, and 8 and D are 

 constants. 



Omitting in the first instance the term in w~ 2 , it is seen that the first approxi- 



Fig. 4. 



mation to z is V~'w, and that to the next degree of approximation the formula is 

 equivalent to 



= (V-8z- 1 )dz, 



whence 



w = Vz-Slogz ........... (31) 



In this expression for w the first term represents uniform flow V in the negative 

 direction of the real axis, and the second term represents a source of strength S at 

 the origin. If there be superposed a uniform motion V in the positive sense of the 

 real axis there results a liquid motion due to the uniform motion of the internal 

 boundary with velocity V, the liquid being "at rest at infinity"; and the first 

 approximation to this motion, for z great, corresponds to a complex velocity and 

 stream function w' consisting of a " source-term," namely, 



w' = -S log z ........... (32) 



Now 2x times the strength of the source represented by the " source-term " must 

 equal the rate at which liquid is being displaced by the intrusion of the internal 



