494? Mr. Lubbock on the Variation of the 



^-da + d £db+ &dc+*£ *< + **.= -*£** (6.) 

 da db dc de dy v ' 



dz' , dz* .. dz' dz dR 



-i—da+ -n-db+— r ~dc+- r - de + &c.= r - dt (7.) 



da db dc de dz v 



Multiplying (2) by ~, (3) by ^-, (4) by ~~ and 



adding the equations so formed to equation 1, after having 



substituted in (1.) the values of - T — , -^ — and —j— from 

 v ax ay dz 



(5), (6) and (7), I obtain the equation 



dR , __ ( dx dx' dx* dx dy dy' dy' dy 



da \ da da da da da da da da 



+ 



dz dz' dz' dz \ , 

 da da da da f 



C dx dx' 

 \~db lU 



( dx 



dx' dx dy di/ dy' dy 

 db da db da db da 



diss dz' dz 1 dz \ , . 



db da ab da y 



dx' dx' dx dy dy' dy' dy 

 ~~da~ dc da dc da dc da 



dz dz' dz' dz "\ r 

 dc da dc da J 

 + &c. 



= (b, a) db + (c 9 a) dc 4- (e, a) de + &c. (I.) 



which is Lagrange's theorem. Again, 



dV _ dV_ dx d£dy_ dV dz 

 da ' dx da dy da dz da 



dx' dx dy' dy dz dz 



dt da dt da dt da 



(8.) 



dt db dt db ^ 



differentiating (8) with regard to b and (9) with regard to a 9 

 and subtracting one result from the other, M. Cauchy obtains 

 the equation 



d (b» a) /T\ 



j. =0 (b, a) = constant. 



a t 



See Liouville's Journal, Oct. 1837, p. 407. 



