54 



Lei <p(^x,tj,a,c,,c.,&c.) = o be the complete iutegra! of 

 (p' [x, y, a, dx, dy, d'y &c.) = o 



then c^,c.., &c. being- arbitrary as to form, it may happen that 

 althousfh the former equation be exact when a is any assigned 

 qur.ntity, yet if its Umit be taken by making a = o, the equa- 

 tion ^Qx,y, c,,c,, &c.) = o will not be the exact integral of the 

 equation <p'Qx, y, dx, dy, d'y &c.). This at first seems to afford 

 an argument against the general accuracy of the method of 

 limits, and a similar circumstance in a case of finite differences 

 has been urged by Lagrange* against the method of limits. 



The objection, ho\vever, lies not against the method of limits, 

 but against the arbitrary form of the constant quantities. This 

 will be readily understood by considering the instances here- 

 after adduced. 



The last is the case referred to by Lagrange. About the 

 year 1782, M. Charles shewed that an equation to finite differ- 

 ences might have two integrals, each having a constant arbi- 

 trary quantity. He applied this reasoning to differential equa- 

 tions and deceived himself as to the result. This result La- 

 grange gets rid of, by attributing- the source of error to the 

 passage from finite to infinitely small quantities. 



" Ainsi il faut dire que passage du fini a I'infiniment petit, 

 " aiieantit non-seulenient les quantites infiniraent petites, mais 

 " encore la constante arbitraire." 

 Let us take the equation 



d"2 + Ad-'-'zdx + Bd''-"-zdx"- -j- - - - Pzd.x'' = o <1) 

 where A, B, C, &c. are constant quantities. 



* Seances des ecoles normales, &c. 1801, p. 401j.&c. 



