6o CALIFORNIA ACADEMY OF SCIENCES. |Proc. 3D Ser. 



or in general 



(2) u GO f _ V d / v U-fl /*',*= 1, 2,   «\ 



(2) ^ ik '-^ a^* v p = i, 2, .. «;• 



§ 2. Relations between the Symbols U Kk 'f. 

 If we wish to investigate questions as to the invariants of 

 the group, we must know how many of the equations 



U& f = ° are independent. It is therefore necessary to 

 find the relations between the different symbols. We find 

 at once 



(3) y i W-V— * c/j^-»/=o 



li = 1, 2, . . n \ 



\k= 1, 2, . . j— 1, J + 1, . . »/' 



i. e., n (n — 1) = n 2 — n relations. Therefore only n of the 

 equations £7 ik (m_1) f = o are independent, and there are 

 obviously not less than n. 

 Further we find 



(4) y t u^-*f-y k u^-*f=y* Uf**f—yi U^f, 



again n 2 — n relations, which show that just n of the equa- 

 tions 6 r ik (m_2) y"= o are independent. 

 Generally we have 



(5) y, U^f-y, cr n V>/=? %Ayy p) y,- 



y, (i - p) *) . 



where the right member can evidently be expressed in 



terms of l/J^^ f, . . U ik {m ~ l) f Therefore, by induction, 

 for every value of p there are precisely ;/ equations of the 

 form U & (P) f = o which are independent. Altogether, there- 

 fore, there are m n independent equations, which form a 

 complete system in m n variables. This system therefore 

 has no solution except f= const. The general group G 

 therefore has no invariants, i. e., it is transitive. 



§ 3. Composition of the Group. Commutator-relations. 



We have 



(6) UP/^X *(.*> 



