609 



be Jx, the usiml formulation of a problem as under consideration, 

 is as follows. ^) 



Given the pn functions of x\ . . . . , .v" : 



vic\ . • . . , vjc" ; k = I, . . : . , p 

 (the contravariant characteristic numbers of the vectors v), and the 

 qn functions : 



'<>/, , »v.i ; /= 1 ,...., 9 



(the covariant characteristic numbers of the vectors w), satisfying the 

 relation : 



1,....» 



_i v]e' Wi, — 



equivalent to ' . 



v/c . w/ = 0, 



we ask, when the system of the total differential-equations 



dx^ . . . . dx" I 



V,' .... v/' = 



«V . . . v/ I 

 equivalent to 



d\ Vi . . . ■ V;, = 

 is perfectly integrable. 



If r and s are two vectors, Ijing in the /^-vector ^,v, and con- 

 sequently satisfying the relations: 



w/.r=:0 , w/.s = , I T= \, , . . ,^ q 

 which is. equivalent to: 



1, ...,/t l,...,n 



2 iviyr^- 1=0 , ^ wi, s^ = , /= 1, , , . ., <y, 



but being otherwise arbitrary, the conditions of integrability are, as 

 known : 



rf-s- = 0; lz=l, . . , ,, q. 

 ox •' da?/" 



These equations are generally covariant') and are equivalent to: 



r^s'^. V^w/=0; I = \, • ■ ■ , q 



1) Cf. e.g. E. VON Weber, Vorlesungen über das PPAFF'sche Problem, pages 93 and f.f. 



dioia dwiv 



2) Owing to the circumstance that the expression -^— ^ — -^ — is generally 



CO variant. 



