650 Mr. F. Tavani on 



Therefore the expression of V through a, c, taken as m 

 system of reference, is 



(1) h'=--a--c 

 Pi Pb 



(2) e '=-l a ->--b--d j 



Pi P2 



s . . on..) 



"j 



and in general 



m—p + l / r \ V 



V'= 2 (- — V ) + — V . . (viii.) 



* ^i V Pm m r p v p l 7 



valid from p = 2 upwards, 



where V denotes the quantity of the general type defined 

 for a, b, c, d.... 



Let us make an important application of the relation (Yiii.). 



P' . 



From a= — we obtain P' = m, then by derivating P' = va 



twice with respect to t, and replacing in the second member 

 to a' its value, 



a' = 6 mod. a' = — 6, 

 Pi 



and to b' its value given by vii. (1), we arrive at 



\ Pi / V Pi Pi J PiPz 



IX. 



We can repeat the derivation of (ix.), and replace to 

 a', //, c' their values given by (viii.), so that in the second 

 member appear only terms consisting of the quantities 

 a, b, c, d, with coefficients which are elementary functions of 



v and the curvatures p±p2ps, If we indicate by <j)(y. p) 



such coefficients, we can write in general 



d n P n 



w» = »^" (r ' />)V '" (x0 



where Y x = a, V 2 = &, V 3 = c . . . . . 



* This equation is given by Prof. Peano in his ''Analisi infinite simale J 

 vol. ii. § 325, by a method Which in this paper I have followed and 

 generalized in order to obtain the relation (viii.). 



t This relation is also given by Prof. Peano, loc. cit. 



