104 Prof. W. E. Story on Partial 



By virtue of (25) and (27), (22) becomes 



p, = */* (. = 1,2,3,...,/,), . . (28) 

 where, of eourse, (23) still holds. From (28) follows 



«*p, = («**. + */*«>"• (* = 1,2,3,...,*) . (29) 

 and, therefore 



x g d In p s = dx s -\-x s du s (s = 1, 2, 3, ... , «), 



so that (19) becomes, when we take account of (4), 



x l du l -f x 2 du 2 + x z du^ + • • • + # rf« = 0, . (30) 



or, by (3), 



x 1 d(u l — uJ + x 2 d(u 2 -uJ + x 3 d(u 8 -v K )+... + x K _ l d(v K _ l — u K ) + dv K = 0, 



. . . (31) 



if we express everything in terms of x v x 2 , x 3 , . . . , x k _ v 



On changing the signs of all terms of (31), distributing 

 —du K equally among the other terms, and adding 



■ (du Y -f du 2 + du 3 + . . > + du K _^ 



to both members of the equation, we obtain 



(— - - *i)d(wi~'0+ (— x - x 2 \d(u 2 -u K ) + U— - x^d(v 3 -u K ) ] 



Putting 



1 

 x. = . — z. or z. = 



-^-1-^ (i=l,*,3,i..,*-l). (33) 



md 



1 



Y(m]+w 2 + w 3 +...H-w k _ 1 ) = "> ? • • (34) 

 taking ^, s 2 , 3 , * . . , z Km . T for new independent variables, in 



