On the Resolution of Equations of the Fifth Degree. 113 



3. From the form which the equation for V first assumes, we 

 see that we must have (p. 564) 



(V-V F ) (V-V G ) (V-V H )x 



(V-VF (a/ 3))(V-V G( «/3))(V-V„ ( ^ ) )x 



(V-V F( « 5 ))(V-VG(«e))(V-V H( * f) ) 

 = 0;" 



Y G and V H denoting what V F becomes when /is changed suc- 

 cessively into g and h. 



Let us examine the five sets of factors which compose the 

 function here presented to us. 



4. Resuming the equation 



and observing that 



we may instantly perceive that the eight functions 



v Fj v f „ 



V F(/3e), Vf '(/?. 6 )' 



will be equal to each other ; (/3e) and (yS) being the comple- 

 mentaiy interchanges relatively to/«. 

 Again, it is clear that the equations 



ff a = ye + Vy + (( J/« 



will furnish a corresponding set of eight equal functions ; the 

 complementary interchanges, which in this case must be taken 

 relatively to g x , being (ye) and (/3S) . 



And a similar result is obtainable from the equations 



V H = P A + P H 8e) n + P* + P*'(*fV 



K=y e +ye+ a iy«- 



We shall thus have twenty-four functions distributed in three 

 groups, consisting each of eight functions. 



Analogous groups must also exist for Vf(«/3), Vg(«/3), Vii(*/3)^ 



for V F (« y ), VG(*y), Vn(*y), . . and for VF(«e), V G(«6), Vii(«e); 



Phil. Mag. S. 4. Vol. 3. No. 10. Feb. 1852. I 



