136 KANSAS UNIVERSITY QUARTERLY. 



,. df,(u,v) df,(u,Y) . df3(u,v) 



<^^) ^x du + ^y du + - du -°' 



,, T,, df,(u, V) dfg(u, V) dfgCu, V) 



(^) ^x dv + ^y dv + ^- dv =°- 



These equations (i) and (2) show that F'^, F'^, F^ are proportional 

 to 



D(u, v) ' D(u, V) ' D(u, v) ' 



abbreviating the work with the help of the notation of functional 



determinants. This supposes one of the three determinants to have 



some other value than zero; if one or two of them have a value of 



zero, the corresponding partial derivatives of F will be zero. 



If they are all of the value of zero, these equations then become all 



alike; that is they reduce to but one equation, and they no longer 



determine F' F' , and F' ; but, whatever 7i and v may be, this can 

 X y z 



take place only when f^, fg, fg are connected by certain relations, by 

 virtue of a known theorem under the head of functional determinants. 



