205 



tinomial coefficients are the coefficients of x* y* 3 . . , j? a/ y p/ . . , . . Let 

 x, y, . . undergo any linear transformation, and let /, m, .., /', m', . . , . . 

 be the coefficients of transformation ; also let 



I m . . 

 I'm'.. 



. 



dl 



dm 



dl' dm' 

 let a a g , a a g , .. be the new values of a a/3 , a a o , ..,; also let 



djc dy 



ay 



then, a une facteur pres, 



and 



will be respectively equal to the combinations of /, TK, . . /', m',.., .. 

 by which the as are multiplied in a a/5 . Again, let/* represent a 

 function similar to/, excepting that in the former the multinomial 

 coefficients are dropped ; then if 



Ono- 



Now the fundamental condition of invariance is expressed by the 

 following equation : 



A" F(a a/3 . . , a a/|3/ . .,..)= F(a a ^. . , a a/j 3,. . 



= F( aj3 .. DajS... aa,p,.. na,^.., ..) U 



where U=F(aa0.., fla,^.., )> 



