12 M. Falk, On the Integration of partial 



they will, in fact, when equated to arbitrary constants, be integrals to (11), 

 (15) and (16). By means of these equations we have, therefore, to de- 

 duce a system of ordinary diiferential equations, to which tlie equations 

 (14), considered as simultaneous, are a solution, f/ and ip being found, 

 an arbitrary relation (f.=j\ii) will be a first integral to (12). The system 

 of ordinary difterential equations , from which (/ and ;// are to be found , 

 may be deduced in the following manner. From (IG) we get 



i=n-\ i=/i-l 



,/,, ^ + o ^n-l-U ^n-ii -^ + O ^n-l-i.i ^n-i.i 



dx '"""' '""'' 



Multiplying in the second member numerator and denominator by 

 ^„_,„ , in the third member similarly by ^„_, „ and subtracting numerator 

 from numerator, denominator from denominator we get by means of (11) 

 and (13) 



^^^^ Yz.^..-yz:-u. h^~^U^> 



(17). 



From the first and second members of (17) we find in virtue of 

 (11) and (13) 



where we have to put successively ^ = 1 , 2 , , . . , vi — 1 . The formula 

 (18) also involves (17), if we, as before, suppose ^„ = 0. 



Now, supposing r=n — 1 in (15), put successively « = 0, 1, 2, 

 ... , n — 1, and having multiplied the equations, thus obtained, in order 

 by /^,1 , ^1 , ... , yw.„_i add them ; we thus get after division by dx 



or, in virtue of (17) and (18), 



^h 



d 



o ^i 'n-i.i 

 ■n-\~i.i <:z:(l 



'dTv Tl 



-^0 



or finally by (12) 



L\ S ^, d:„_,_„ = Vdx (19), 



