18 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



What is required is the elimination of the functional forms from these equations, 

 if this be possible. 



Manifestly all the third derivatives of < and disappear in the combination 

 a + / g h ; in fact, 



If, by means of the expressions for , w, , it be possible to eliminate 6 and (/>, we 

 must have a relation of the form 



+/- 9 ~ h + & + V n + & = > 



where , r;, , do not involve 6 or <. 



In order that the terms in ^ and < 3 may disappear, we have 



'rt + (1 + c',) g= 0; 



that those in <j> n , ^> ]3 , <^ 23 , may disappear, we have 



= 0, - (a\ -- 6'j) + <? = 0, 



that those in L and 0. 2 may disappear, we have 



(1 + a) f + #, + (1 + y) ^ = 0, '| + (l + ^8') , + y '^ = ; 



and, finally, that those in n , 12 , ^ 22 , may disappear, we have 



- y + M?+ ) = o, - (*' - y') + ^ = 0, 



- ( - r) + (' - y') + /* (f + ) + M = o. 



These equations are to be satisfied simultaneously. 



Using the last set of three, we have, on substituting in the third from the first and 

 second, 



and similarly, from the first set of three, 



(p + cr 

 We accordingly take 



