ON THE VORTEX THEORY OF THE LUMINIFEROUS ^THEE. 491 



find, -with V^, as in §§ 8, 9, to denote the average i/'Component-velocity 

 of the turbulent motion, 



A(xzav(ui;)} = -V2fe^"^-Q . . . (36), 



-where 



Q=xzav< \x-\--' + V — — - +u- +v — +" T" > • K^O- 



I dx dy dz ilx dij J 



18. Let 



_p=)J + 'cr . ... (38), 



■where ^) denotes what p -n-oald be if/ were zero. We find, by (5), 



_^',^=^.fiyA 1^ . . . . (39), 



dtj dx 

 and, by (27) and (37), 



Q=zzaY(/^+u^!'^) .... (40), 

 \ dx dij I 



So far we have not used either the supposition of initial isotropy for 

 the turbulent motion, or of the infinitesimalness of df\dij. We now must 

 introduce and use both suppositions. 



19. To facilitate the integration of (39), we now use our supposition 



that - /(y,0» divided by the greatest value of /(?/,0> is infinitely small 

 in comparison with »i, n, q, which, as is easily proved, gives 



^^^MvA „L-J^. . . . (41), 



dy — V dx 

 by which (40) becomes 



Q^_oJ^(M.zav(.;L + «^)v-^^. . (42). 

 dy \ dx dy] dx 



Now, by (a*, s) isotropy, we have 



n ! d , d\ „_o d^n 



2xzav{i'o-p + Uo r)^ ^ 



\ dx dyj dx 



Performing integrations by parts for the last two terms of the second 

 member, and using (1), we find 



dx ^dzjdy \dx dz J dy 



dVn d _, 



= xzav — - — V -Vq ; 

 dy dy 



and so we find, by (43) and (42), 



20. Using now the Fourier expansion (7) for Vq, we find 



4-4 ^^v< o f-./.f^'f^oa (mx + p^ sin (ny+f ) cofi(qz-¥(i) (a^^s 

 — A -»o=222222p, , —5-- — 5-; — s K^'^J- 



