140 ON THE SOLUTION OF 



This relation shows that ^r'x is constant, for a series of points 

 in the curve, and therefore, Mr Babbage reasons, we may con- 

 sider it as a constant in (16), which thus becomes an equation 

 in finite differences. He integrates it on this supposition, and 

 adds an arbitrary function of ty'x, which has been treated as an 

 absolute constant. The result is therefore 



which is an ordinary differential equation. 



This process appears to have been suggested by an incorrect 

 analogy with the way in which arbitrary functions are intro- 

 duced into partial differential equations. 



A little consideration would have convinced Mr Babbage, 

 that by integrating (16) as an equation in finite differences, he 

 only passed discontinuously from one ordinate of the curve to 

 another, and therefore could not obtain a continuous relation 

 between x and y. The legitimate result of his process is 

 merely, 



where n is any positive or negative integer. This is quite 

 different from 



In exemplifying equation (20), Mr Babbage first supposes 



dy\ dy 



-f-\ =00 x y -r- + 



dxj * ax 



and thus obtains the equation of a straight line parallel to Ox, 

 as a solution of the problem, which undoubtedly it is. 



In his next example fly ~j = a 2 . By making the con- 

 stant of integration imaginary, he gets # 2 = a* x 2 , the equation 

 to a circle. But although this is also a real solution, it has 

 no connection with the relation -^r [x + ^ yfr'x] = ^'x, from which 

 it appears to be derived. It is, as we have seen, a particular 

 case of the complete integral. Consequently if the method 



