60 



same reasoning, which appHes to tlie integration ot equations of 

 finite differences, when apphed to the Hmits of these equations, viz. 

 to differential equations, leads to error. The abo\'e integration is, 

 as to substance, the same as in the work of Lagrange abo\'e re- 

 ferred to. He attributes the error to the passage from finite quan- 

 tities to indefinitely small quantities, and thence to the limits. 



But this appears to me a mistaken view of the question : the 

 second value of a (14) is not as general as it ought to be, and 

 cannot apply to the limiting equations, for in equation (13) we as- 

 sumed u=br' and then u=br'+', now b being any arbitrary con- 

 stant quantity if we make as in the case of the limits i = o, u—br" 

 and u = bf and therefore, u — u, but this is impossible for by equat. 

 (12) u + u-o an equation that cannot subsist u and u being equal 

 and finite quantities. Consequently that the values of u and w 

 may apply to the limits, it is necessary that the constant arbi- 

 trary quantity should be of the form i b instead of b, then when 

 we take the limits by making i=o, u~o, and u—o and therefore 



If we carry on the process with u—ibr'', the second value of 



.r 



a =1 b ( — 1) ' — -^ — -J, b being any constant arbitrary quantity. 

 So that one of the integrals of the equation 



^ ^ .rA, ^ LM' is 



y 



- ax -\- u- 



where a = ib (—1) ' — ^ — ^, b being any constant arbitrary 



quantity. Here evidently i b is arbitrary as to quantity and may 

 be represented by b for all values of * excepting i~o. 



