290 CELESTIAL MECHANICS. I. vii. 32. 



supposition that h and c only vary, while a, t, and z remain 

 constant. Hence 



dV - dV ^ d'z . d'2 



do dc do dc 



d'z / dV dTz) 



d6 , , JdV dc'd6 

 consequently dc= — -jt- d6, and DyzzDol-Tr 



= Dft, 



dVd^___£y£f 

 d6 dc dc *d6 



dj^ 



dc 



£z 

 dc 



djz 

 dc 



: which is the height of the paral- 



lelogram (x). Its base is equal to the section of the pa- 

 rallelogram formed by a line parallel to x, belonging to a 

 plane in which those particles of the parallelepiped {A) are 

 found , with respect to which z and y are constant : the 

 length of this section is therefore equal to the element of 

 X, supposing z, y, and t to be constant. 



We have therefore, for the element Da:, the three equa- 

 tions 



d'a: d'jT - d'a: 



Da:=-7- Da-f -77- Do-f -r— Dc 

 da do dc 



A dV d'y , . d'y ^ d'z d's , d'2 



0=-r^ Da+ -rf D64- T^ DC; 0=-— Da+ — d& + t- dc: 

 da do dc da do dc 



d'2 



dV 



[and multiplying the second by — , and the third by — ^,we 



have 



^ d'yd'z d'yd'z , d'y d'z d'y d'z , , 



0=-r^.T-Da+ -^,-r- lyb—r^,— Da— =-^.— DO: whence 

 da dc do dc dc da dc do 



Dbzz 



d> dz_dy d^ 

 da dc dc da 



d'y d'z d'y d'z* 



dc 'db db 'dc 



Da ; and in a similar manner we obtain 



