Prof. Clausius on the Conduction of Heat by Gases, 521 

 Thus we have 



IR=>i/2Vl— co&4(u+%S-iqfie)*, . . . (39) 



which product must be introduced into (33) and the integration 

 then carried out. 



For this purpose we require to know what ratio cos c£ bears to 

 the cosine denoted by /a. We have used jju to stand for the 

 cosine of the angle formed by the moving direction of any molecule 

 whatever with the axis of x, and <£ to stand for the angle con- 

 tained between the direction of this molecule and that of the given 

 molecule. Further, let the angle which the moving direction of 

 the given molecule forms with the axis of x be <n, and the angle 

 between two planes passing through the moving direction of the • 

 given molecule, and containing respectively the angles <f> and ??, 

 be -f. Then 



^ = cos77COS<£-f sin ?; sin <£ cos tjr . . . . (40) 



The superficial element dco may at the same time be represented 

 by sin (pd^d-fr. The equation (33) thus becomes 



R= ^— \ \ dcpdty sin <f> V'l — cos <j>[u + ±& 



— ^(cos7?cos<£-{-sin77 sin<£ cos^)e], . . (41) 



where the integration according to ^ must be carried out from 

 to 27r, and that according to cj> from to 7r. 

 By performing this integration we get 



1=^ + 18 + ^ cos 77. e)f. . . . (IX.) 



§ 20. We must introduce this expression for R into the equa- 

 tions (31) and (32), in order to obtain the values of a and «. 



* [Thus in the original : probably a misprint for 



IR = V2Vl-cos<//w-h 2«- o^7~ I V2V1-cos</>(«+28-2?pOJ-/". 



— G. C. F.] 



t For the sake of greater clearness, I have omitted in the above calcu- 

 lations all terms containing any power of e higher than the first. I will, 

 however, here give the result of the more extended calculation in which 

 terms containing the second power of jf are also included ; namely, 



= 4f ,U, 1 .IS 2 1 Be 



R =3r + 2 8+ I0 9COS,?e+ 4^-5 (?COS V 



