Leathem — On Two- Dimensional Fluid Motion. 



37 



For S real and a > $ > 0, 

 and therefore, at £„, since <7 = | l/^'(£ ) 



j = o 



log 2 = log 



Jr (» i +(ff-g,) , |"' 



+ 



1 



7T 

 = 



(q-E)*-(«-W* 



(q-g)*.+ («-£,)* 



(64) 



If 5j be the change in 3 due to the S^' above specified, it follows from 

 formula (64) that 



K = y 

 $q \dd , 

 — = — log 



(a -g,-A)i -(«-£„) * 

 (a-5,-X)*+(a-5.) i 



</0(A) , A 



- — 1 Incf 



tt 10 ° 4 (« - L) 



the last equality being approximate, the second power of A being neglected 

 under the logarithmic sign. Thus 



9. 



log |4 («-£,)) 



(7 0(A) 



1 



7T 



log|A|<7 0(A), (65) 



-7 - y 



in which it is to be noticed that the first integral, by (63), equals %%', and 

 that the infinity at A = in the second integral is not sufficiently powerful 

 to interfere with convergence. 



It may be assumed that 6 (A) is expansible in the form A + B\ + CX 2 + , 



so that d.6 = {B + 2CA + )d\\ then 



7 7 



log I A I dd = {B + 2CA) log I A I dX, = 25 (y log y - 7), 

 -7 - 7 



approximately. Also, by (63), 2yB = $%' approximately. Therefore the 

 second term of (65) equals (Sx'/^i^ogy-l), and 



Sj _ S x ' 

 2 



This formula represents the change in q at a particular point due solely 

 to a variation S\ distributed in the immediate neighbourhood of that point. 

 If account has to be taken at £ = $,, of a variation of d\ elsewhere, 

 say Six' at £ = &, there is a further variation of q, namely given by 



= §jf |log(J8fJ-log{4(«-f.)J-lj. 



m) 



Si? .. ?i'x 



log 



(q - gQ* - (a - g,)* 



(« - ft)* + C« - So) 1 



(67) 



