0 
l—s w u en 
uw = — ef se V1— BF sin’? w dw, 
Vw, w,tw, t 
‘am 
w, 
w, + w, 
to the limits of the integrals evidently zis also = O for y = O, hence 
wy ='/,~7. And as at the upper limit w,? + w, becomes = 0 (culmi- 
when =k? is put (hence 4? is always <1). With regard 
nation point of the collision), also (1 — 2”) (1 + ze )=0 hence 
z=1,y=0. Thus we have, after reversal of the limits, in conse- 
quence of which the minus signs drop out: 
Ig 7 
sae frm vbw. ay 
—s mu a dw ats 0 
= jie Vw cians Ay pee te Ip 7 ; 
Ly 1 10 w 1 {x 
Aw 
0 
when for ¢ its value is substituted in the expression for u,?. Follow- 
ing LrGENDRE and denoting the complete elliptical integral of the 
Yor 
d 
1st kind, vin. [= by #’, or singly F, we have also: 
0 
Ig 7 
2 
Pp m w, + W,U es 
t = —______ — FF, >= — — Tw Aw.dw, 
les el iy, > F fo w Aw.dw 
0 
m l—s 
2f is substituted for —— (see above). We have for the 
; u 
when pV 
lo 
modulus 4: 
a es ee ‘1,  [p(l—a) + Y p*?(1—a)* + 2a] . 
WW, p Vp(l—e)' + Ze 
when we calculate the quantities w, and w, + w, from the above 
expressions for w,—w, and w,w,. Hence we may write: 
1 1 
kid MA ver Lj 
2 va 2a 
1 + —_— 
p* (l—a)’ 
so that at low temperatures (large values of @) £* is always near 
1 (provided a <1, in which case the + sign is valid). 
We must now still reduce the last elliptical integral (the one with 
