Royal Society. 385 



A = I I m . . j 

 \ I'm! .. 



>'= l 'jl + < + - [ 



. ..J 

 let V . • a «, P , ' • • be the new values of « a8 , a , , . . • also let 



t-7 _7 & , rf , 



flte ay 



ax dy 

 then, a une facteur pres, 



and 



<^ a »n o*a „ 



will be respectively equal to the combinations of l,m, .. V, m\ ..... 

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

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

 coefficients are dropped ; then if 



U m ..=f(l,m i .._A_ ) d \ 



\ da «p.. da «,?,.. ' 



1.2..(n- a ).l.2.../3.. a "°-- 

 Now the fundamental condition of invariance is expressed by the 

 following equation : — 



mn 



A* F(a^.., a ai$i .,, ..)=F(a^.. ( a „ A ), 



=F(a«/3.. n^.., a« A .. U aSi ,,, ..)U, 

 where U=F(a a/i .., a xSi .., ..), 



and the problem consists in determining the coefficients of U This 

 gives 



A*(«0./.« 1 ft..''...)=n^.nJ A ....U, 



the quantity within brackets being the coefficient of 



ftp. . OLjfif. . 



Then if the term be selected in which i=i l= =.. = \ f an d if A~»" be 

 »o transformed that it shall appear as a function linear in each of 

 Phil. May, S. 4. Vol. 9. No. 60. May 1855. 2 C 



