314 



30 



il constant, wl = w^t + âk, to^ = (|^) , (k = 1,2,3) (84) 



where we have denoted by (^y-j the value of , obtained by introducing for 



/j, /j, /g the vahies 1°, II, /° respectively. In order to find the /^'s and w^'s let us 



first consider the first three of the equations (81). As does not depend on the 

 iv's they may be written in the form 



1^ = -"^. (i. = 1,2,3) 



dt div,. ^ ' ' / 



The right sides of these equations are, as seen from (82), functions of the /'s and 

 if's, but if in the calculation of the / ^'s we neglect second and higher powei-s of F 

 we may for the /'s and w's introduce the values for I'l and w\ given by (84), so 



that the differential coefficients become equal to known functions of the time. 

 Neglecting for simplicity, here as well as in the following, the constants appearing 

 in (84), this gives 



where the quantities denote the expressions obtained by replacing in the quan- 

 tities Dt the /'s by the /°'s. These equations may be directly integrated and give, 

 if the arbitrary constants are chosen such that in the expressions for the /^'s no 

 constant terms appear, 



I\ ^ - eFl'= tl7^^^^ cos27r{V^^ CO, 4- (o,)t, 



<üj -|- CO 



1\ = — eFZ cos27r(r- \w^ + cv.}t, 



~ 1 ft»! + 



= 0. 



(85) 



Among the terms on the right side of each of these equations the term correspond- 

 ing to r = is much larger than the other terms because for r = the denomi- 

 nator (r— 1)^1 + ^2 becomes equal to — w^ + Wa' ^"^ this quantity, which will 

 be denoted by o, is a small quantity of the order /. In fact, from (84) and (76) we have 



U/7j«+U/J - [ c I p{ii-^nr 



The term in (85) corresponding to r = becomes therefore of the order Z^/;., and 

 we may according to what has been said in the beginning of this section neglect 



