902 
sinty) to that of the 1st and the 2rd kind. According to known 
formulae of reduction') we have: 
D 
: 1—k dp 2k?—1 
for wAw.dp= | fz “+ aA dp sin eos pAw | 
0 0 
hence 
Yom 
eeu 1fl1—k?  2k?—1 
f si WA w.dw = 5 Dek + a E |, 
0 
when the complete elliptical integral of the second kind, viz. 
Ip 7 
Jere is represented by Z. Hence we find finally : 
en en ee RA ee ; 
pe Pope SO oe Le eee es 
because 
p 2k*?—1 w, 
mn = —, and kn 
Vw, +w, VY p(1—a)? + 2a l—a w,-+w, 
We shall now compare the found formulae (7) with those found 
before, and again in the two limiting cases: high and low tempe- 
ratures. 
At high temperatures (p small) £7 approaches '/, (if a < 1), so that 
then (7) reduces to 
B=") ys epee ws? (lagh temp.);— “or ee 
instead of w,2—7'/,u,? (weak collisions), as we found before. The 
1) See among others Duriee, Th. der ellipt. Funct., p. 65, formule (29), i. e. (with 
u =0,m =0) 
sin Wp cos A yy fi _—_— 2 (1 nf" en : Ee vn fe ae = 1 
from which the Lanett with sin*) can she expressed in an the others, that 
with sin? being expressed in Fy and Ey by the formula (see p. 69) 
Le 
„ (ae wap ri 
Aw 
0 
as can be easily verified by differentiation, after 2v=W1—k?sin*) has been 
put everywhere in the denominator. (ne integral to be reduced by us then becomes 
y 
{> wy — k? sint p aw). 
Aw 
0 
