FUNCTIONAL DIFFERENTIAL EQUATIONS. 137 



In (12) we may write for %c 



Hence % 2 a;= ffi^x &c. =&c., and (12) becomes 



= .................. (14), 



by putting <j>x for x f being a known function, (14) is a 

 functional equation of the first order. 



Such is Mr Babbage's method. Let fx = 1 4- x ; (14) be- 

 comes 



and if we denote $x by u x , and replace x by 2, we shall ob- 

 tain (13). 



It must be admitted, that it is difficult to prove that the 

 generality of (12) is not restricted by these transformations. 

 They are however often useful, and serve to illustrate what 

 was remarked in the last number, with respect to the affinity of 

 functional equations, and equations in finite differences. 



If, instead of (10), we had taken the more general equation 



where A is an arbitrary constant, precisely the same method 

 would have applied. In this case the factor ty"x would 

 have led to the result 



tyx = a.x 4- ft, 



and by substitution fB = a + /3 + A, 



therefore a'+A = Q, or /8= oo . 



Now in the case we have been considering, the former con- 

 dition is not fulfilled ; hence we must have ft = co , and the 

 geometrical interpretation of the complete integral is a right 

 line at an infinite distance from the axis of abscissa?. 



We not unfrequently meet with similar cases, in which the 

 complete integral becomes nugatory or impossible in the pro- 

 cess of introducing the necessary relation between its constants. 

 Under particular conditions, however, this difficulty does not 

 occur, and then we obtain, what in the ordinary methods of 

 discussing functional differential equations, appears to be a 

 conjugate solution, unconnected with any other ; (15) would 

 be an instance of this, were a + A = 0. 



