4 S. Chapman — Kinetic Theory of Gases 



where 9 represents the inteerrand as defined by the radical 

 expression in brackets | | "^ and fe denotes j cpc^^, which is a 



function of y^^ and n. *' 



The integral for 6 is now in a form suitable for numerical 

 computation, because 9 is finite throughout the range of 

 integration, and has a finite limit as z -^ o or A -> i. This 

 limit is readily found to be given by 



(3.9) Lt cp-xo ((n- i)coixo + 2 ian lA= sino^' ^^^ + '''' ^^0} 



the second form being the more suitable for use in computa- 

 tion. 



This gives the integrand of (3.8) when s = o. When ^=i, 

 or X==o, evidently 9=1. For intermediate values of z the 

 value of 9 must be found by direct calculation (cf. §5). 

 In this way, bv applying Simpson's or Weddle's rule, can 

 be found as a numerical multiple of y^^ the factor itself depend- 

 ing on xo. 



(4) In the integrals Ii and I2 it is convenient to change 

 the variable from a to y^^ which is defined as a function 

 of a by (3.4). This renders the range of the transformed 

 integral finite, since as a varies from o to (3o, y^ ranges from o 

 to \'K. Also 



n — 3/ 2 \Ji- tan\(, /n-\-i \ 



(4.i)a da = \d a' = \^\^'^-' "1^(^-3 + ^^^ 2x0)^X0 



{cosxqY-^ 

 so that 



(4.2) I,(n) = /00P"co.^2/.y., ^^^(^+...2X0)^X0 



and (-%r- 



(4.3) 



L(n)-/(?i) I ^'"^ sin'-2ky,.cos-^2kx, — ^/^Xo^ Cj^l ^^^-^ ^XoJ^^Jio 



where 



(44) / {n)=2T: ^^^^(-^^ Y"^i 



The integrand in (4.2), (4.3), vanishes when Xo = 0- When 

 Xo = ix, in the denominator (allowing for the factor tan x^) there 

 is (cos^o)^'^"^^''^^'''^^ ^^'hile in the numerator the vanishing factor 

 is cos'^zkyoj in which k tends to i as -/o tends to 1%; hence 



