59 



therefore r r: ( — l)f 

 andM = 5( — 1)T 



X 



Hence a = b (— 1) ' — t — T (14) 



where i is a constant arbitrary quantity. 



consequently we have obtained two integrals of the equation 



viz. y=ax+a^ where a is any constant arbitrary quantity, 

 and y=ax+d' where 



X 



X 



a is any quantity of the form b ( — 1) ' — "2 "" "S' 

 b being any constant arbitrary quantity. 



If we now suppose i to be diminished, and the limit ot 



■^ to be taken, i. e. |^ be substituted, equation (14) becomes 

 y = ''j£ + (Ml (16) 



Now the integral of this equation is as easily appears y=ax+a* 

 a being a constant arbitrary quantity. This coincides with the 

 first integral of equation (10). The second integral of this equa- 

 tion becomes in this case, because i is evanescent and because 



2x X 



y = b'— 1 (17) 



According to this conclusion we have two integral equations 



each containing an arbitrary quantity for the same differential 



equation of the first degree. This cannot be. And in fact the 



equation ("17) will not answer. Thus it would appear that tlie 



K 2 



