284 



CELESTIAL MECHANICS. I. vH. 32. 



6 F 



" The coordinates, x, y, z, of the point M, become, at the 

 end of the instant Af, a: H-wd^, y + vAf, and z + m^a^, which 

 are therefore the coordinates of the point M', and those of 

 any other angular point may be found by substituting the 

 corresponding variations : thus for the point N', the ordi- 

 nates are at first x, y, and z-j-Dz, and afterwards, u being 

 changed to m 4- D m in each instance, we have for the 



d'M 



d'w 



new ordinates x + utd + -r- DZAf ; y + viM -f — r- DZAf , 



ciz dz 



and 2 + Dz 4- it?Af H — r— dza^. The differences are -— DZAf , 

 dz dz 



dza;, and DzH — ; — DzAf, and the sum of their 



dz dz 



squares will be the square of M' N' : but the two former 



being infinitely small in comparison with the latter, their 



d w 

 squares may be neglected, and M'N'zzdz+-- — DzDf. 



dz 



** The coordinates of the point E' must be deduced 

 from those of M', and the coordinates of F' from those of 

 N', by substituting x-\-\ix and y + Dy in the place of ^ and 

 y : consequently the length of E' F' may be deduced in the 

 same manner from that of M' N' ; hence we have 



