•^o Carnoi on the TJjcory rf 



60. Othcrwife : Let us confider the propofecl cutvc as ll 

 polygon of an infinite number of fides; that is, inllead of the 

 curve, let us take a polygon of any number of fides whatever, 

 and let us then fuppofe that number of fides to Increafe con- 

 tinually; fo that tlie ultimate relation of the polygon to the 

 curve may be a relation of identity*. As it is abfolutely 

 impoflible that the curve can be accurately confidered as a 

 polygon, the equations by which I fliall exprefs the condi- 

 tions of the problem depending on that hypothefis will not 

 he exa(ft. But, fincc the polygon is fuppofed to approach con- 

 tinually to the curve, ihe errors, which may exift in thefe 

 equations, may be diminifhed as much as we pledfc; and hence 

 thefe equations will be what I call imperfe6t equations. 



Thus the triangles T'MP and MNO give the equation 

 Tfp MO 

 ifp ~ '\7T'' ^"^ fubrtituting TP for T'P, which differs 



infinitely little from it, we fhall have this imperfe6l equation 

 TP MO TP dx 



"Wp ~ IvrT' ^^ — ~ "7^' ^^^^ (dmc with that before 



found, and which, being combined- with the equation of th^ 

 curve, will give the fame refult, 



61. Again: We may apply to this problem the method 

 of indeterminates, without making any alteration in the pro- 

 cefs. After finding the two imperfect equations, 



— - = —j—y and Zj'ay = ['zad.v — zxax), 



I add mentally the quantity <[> to one of the fides of the firfl: 

 equation, in order to render it rigoroully exa6l ; and I intro- 

 duce, in like manner, into the fecond, a quantity <C>', which 

 renders it equally accurate. The quantities thus underilood, 

 namely $ and ^', are therefore infinitely fmall compared 

 with thofe to which they arc mentally added. This being- 

 fettled, I compare the two preceding et]uations, without 

 having any regard to the quantities^ and <[>^ Bttt as the 



equation refulting from this equation, namely, TP = -rj- 



yy 

 X ■ ■■, may not be exaft, I again add mentally a quan- 



'^ I-#ok back to the fecond note on article 42. — W. D. 



' ' tity 



