128 Mr. li. Moon on the Transmissim 3 in one 



Hence we may take for the auxiliary equations, 

 0-df v 



Wo - w l 



= dx — ae 2a . dt } 

 = dco o 



CO 



'A = 



2a 



(15) 



(since by (12) we have log. a x — — ~ — -), the integrals of which 



are 



© 2 =C". 



Hence, putting m for the equivalent of C, we get for the value 

 of/ 2 derived from the integration of (14), 



or, since /j = 5 



On the other hand, equation (13) may be written 



o= -Tf/i'M-Z/M "U'W -f/iW/i'M +//K)/,'W- (is) 



1. Hence, if this last be a substantive equation between the 

 partial differential coefficients of/ 2 , we shall have for its inte- 

 gration, following a method precisely similar to that already 

 adopted, the auxiliary equations 



Q=df v 



(0 2~ ft) l 



— dx-\-ae ** dt, 

 = dco l ; 

 and, as before, we shall arrive at a solution of the form 



j£ =/«'{»* "li- 

 lt is clear, however, that if we have 



/ 2 = funct. (m, &> 2 ) 

 and also 



/ 2 = funct. (rc, «,), 

 we must have 



m— funct. (rc, o^) 

 and 



© 2 = funct. (?z, ©J, 



* Note that in the integration of (15) we shall have w 2 constant; so 

 that by means of the equation / X = C we may express &>j in terms of x and 

 t and of quantities which are to be regarded as constant. Thus (15) will 

 always be integiable by means of a factor /u. 



