126 



Mr. H. Moon on the Transmission} in one 



Eliminating -j — -- between the first of equations (4) and the 



last of (9) a and between the last of (4) and the first of (9), we 

 get 



d*a 



d*cc 



the addition of which equations gives 



Q-F^FiW -F'aWF', (t>) + F' 2 (,)F l (/) -F,(*)F,(*,). (10). 



Again, eliminating -77-77 between the first of each of the pairs 

 (4) and (9), and also between the last of each of the same pairs, 



d*a 



a«F,WFiW«F i W^iW^{PsW^iW^nt^i(«*)}a5 



0=F a (i;)FVa ? )-F 8 WF 1 (f;) + {F 8 («)F I («J-F 8 («,)F 1 ( w )}|^. 



Hence, if (i) holds, we shall ha 

 the last two equations and (1), 



d 2 a d^u 

 Hence, if (i) holds, we shall have, eliminating -j^» -j-^ between 



o=P 8 WF I (o-r , ,WF l W+^{F s («)P l (*)-P,(*)P l («)}.(U) 



Equations (10) and (11) express the conditions to be satisfied 

 by Fj, F 2 in order that (1) may be derivable from a combination 

 of the derivatives of (3) . Transform these equations by putting 



Y 2 (xtva x ) ss/ 8 (a?/»!.fflg)j 



^v + alog.*^ 

 > 2 =tf— «log e a/J 

 then we shall have 



where 



a). 



*YM= j {/VM-//K)}, 



F,'K)=f{/ 2 'W-/ 2 'K)}^ 



Substituting these values in (10) and (11), they become 



