314 LIPKA. 



,„ . dFi Pi dFj^i dFk Pk dFk-x /• 7 1 o ^ 



(/5j = (?, A: = 1— > 2— ♦. . . -^ n). 



dpi pi_i dpi dpk pk-i dpk 



If /c ± i + 1, we may replace — ^ by —^ —, and (75) becomes 



dpi pk-i dpi 



dFi Pi dFk-i ^ dFk Pk dFk-i 

 dpi pk-i dpi dpk pk-\ dpk 



and by subtraction of Fk-\ from both members, 



— - {pk-i Fi — piFk-i) = — - (pk-i Fk — Pk Fk-i), 

 dpi dpk 



or 



(76) a«^^^aa^^. {k^i-^D. 



dpi dpk 



If A; = ?' + 1, we may replace ^ by —^ ^, and subtracting 



dpk ' 2^1-1 djik 



Fi-i from both members, (75) becomes 



(^0 —z = -^ , (A- = ? + 1). 



dpi dpk 



From (76) and (77) and with the help of (74), we may now write the 

 series of conditions on the as: 



dpi dps ' dpo dpi 



dai+i, i doik ^ dajk , dak, k+\ _ ^. 



dpi+i dpk dpi dpkti 



Hence the a's are given by 



(79) aik = pk 0i — pi 4>k, (i, k = 1, 2, . . . , w) 



where the 0's are arbitrary functions of the coordinates .ti, .To, . . ., .t„ 

 only. 



Combining (79) and (73), we must have 



