1898.] with two-dimensional fluid motion. 391 



where k ly h^ k 2 , h, 2 , ... are real positive quantities in ascending 

 order of magnitude. It is then found that, for positive integral n, 

 we have 



where o^ 1 - /^ty? = I, and 







gr-u ^ ^ r + 



2 P lWA r -VV WA r + vV 



the square roots VA r , V&,. being taken positively ; here e r , = + 1, is 

 a quantity which is not determined by the positions of the circles, 

 but must be assigned when the substitution S- r is put into the 

 form 



with a w S (r) - yS<V r) = 1- 



From these it immediately follows that if 2; be a pure imaginary 

 quantity, and z' = — z be its conjugate complex, y% ] z + S^ is real 

 and equal to yllj,^ + & { l! n , while a^z + fi ( ^ is purely imaginary and 

 equal to the negative of a ( !_ ] n z' + $ { ? n . Therefore if 



f(0=...V 1 ^- B9 ,..(D=^f , 



be two substitutions in which the factors formed from the primary 

 substitutions follow one another in the same order, but for any 

 factor ^" in the first there enters, in the second, the factor ^r~ n , it 

 follows, if z and z , = — z } are conjugate complex quantities, that 

 <}> (z) = — <f)' (z') t and <yz + 8 = y'z' + 8'. These facts are clear 

 enough from a consideration of the figure. For our purpose 

 another fact is also necessary, namely 



when e r = + 1, we obtain 



O«0r + /3 ( ''> = - C r = C,', yWOr + S< r > = 1, 



when e,. = — 1, we obtain 



a "■' a,. + 13 « = - a r = a/, 7 w a r + 8 {r) = 1 , 



where c/, a r ' denote the conjugate complexes of c r and a r re- 

 spectively. 



