statistical meclianics 



But 



= COj^ COS XIO^ , 

 = COS Î/Ci)^ , 



tv^ = (0^ COS zio^ . 



As further 



cos Xb>^ cos + cos yiü^ cos i/w + cos zio-^ cos = cos (ii^n, 

 the right member of the above equation therefore takes the form 



jda de^f^ Q co^ cos ww^ ^ j'^^'^ ^'i/i Q G'n, 



o G 



where Gn denotes the velocitj^ component perpendicular (outwards) to the limiting 

 surface. 



But do Gr„ = an element (dii) of the space between the limiting surface at the 

 time t and the limiting surface at the time t + dt. Consequently this integral is 

 nothing but the integral expressed above as 



jf\Qd^, dii, 



so that 



(2) + 



or otherwise written 



(59) 



Q f^ ds^ dÜ 



Ai2 



Mi 



/, \ u, \- V. \-w. — de, dii 



From this equation, together with (57), we get the following general theorem. 

 Let 4> be any function of t; x, y, u, v, iv. Then (supposing the integration regard- 

 ing M, V, tv to be performed between constant limits) : 



(60) 



dj^ds dü 

 Ü 



dt v 



dB dil 



o 



84^ 8<1> , 3<î> , d<t> 



— \ u -! v -j- — W 



dt dx dy ds 



a theorem that might suitably have been placed at the head of our demonstration. 



28. Considering now (3) we get (the forces being independent of the velocities) 



'df\X , d.t\Y df\Z\ 



(3) = - 



du\ 



Q ds^ d9. 



j\ dn^ dv^ dw, I ' 



+ 



du^ dv^ ■ ^ dwj 



