55 



Now it does not immediately appear, that the remit from 

 tAiis solution should be the same as the erroneous result of 

 Newton; but a little consideration will make it apparent. 

 fg is made to disappear in both solutions, by the substitution 

 of FG, and therefore a common error might be expected in 

 each solution. This is proved as follows: let y represent 

 the subtense of the arc «+/>. Then FG — M ox ^.will be 

 •y-yi:^)" or -''>^>''' + ^;:|^y + '^-^>^ This latter, quantity, 

 therefore, cannot become ultimately (^-^, unless « (7 — 7') 

 vanishes in respect to 47/). Now it may be readily. shewn, 

 in^ manner similar to that in which the limiting ratio cxf 

 ^fg—^kl : ^¥G—^kl was obtained, that ultimo a.(y—y')= 

 -^yp. It is easy to see that p"{y—y') vanishes in respect to 

 "'(7 — rO- But that »{y—y') vanishes in respect 4yp is only 

 an assumption. 



The other solution of- Lagrange, in which the same con- 

 clusion is deduced, is the following. 



" La solution de Nmvton pent 6tre rendue plus simple et 

 "*' plus directe de la maniere suivante. En nommant u la 

 " Vitesse dans un point quelconque de la courbe, ?/^ est 

 " Tespace que le corps parcourait sur la tangente dans le 

 " temps 6, en faisant abstraction de la gravite ct de la 

 " r6sistance. Nommont g la force absolue de la gravite, 

 " et r celle de la resistance, ^ et ^ seront les espaces par- 

 " courus, en vertu de ces forces, dans le meme temps 0; 

 " ainsi le corps aura parcouru, suivant la tangente, la ligne 

 " «^ — V' ^^ suivant Tordonnee y, la ligne ?^, dans une 

 " direction contraire k celle suivant hiquelle cette coor- 



donnee 



