558 Mr. G. B. Jerrard's Reflections on the "Resolution 



volved in any of these four equations are either^^,^, ovf^^f' 

 the indices being in the former case equal to the elements of 

 the given affix (^"/^}; and in the latter to those t>f T"/). 

 Further, we obtain 



which involves the single function^, the index of which, 8, 

 does not enter into either Q^^^ or T^y). 



We are thus conducted to the equation (20.), 



/a(«?)=A(rf> ..... (f.) 



where a and b are such as, abstractedly of the order in which 

 they are arranged, to be restricted to the three sets of values 



;}' 



71 ^ 



24. When « instead of occurring among the elements to be 

 interchanged remains fixed, we have 



/b(10=/e(r.O' (g-) 



b and e here depending on 



n r\ n 



6 / » 8 J' «/• 

 This theorem may either be derived from the preceding 

 one, or obtained directly from the equations (d.). 



25. Finally, we obtain 



'/b(^.r)(lO==(^/c)(/3.f)«;. . . . (h.) 



where (^/^^)„ must coincide either with (^^^ or widi (^7^), 

 the complementary interchange relatively toy^. Wllh respect 

 to b and c, if we take b successively equal to 



«j ^> y, % h 



the corresponding values of c will be, if (^ s^ ^ = ^^ s^ 

 but if (fj^n = (yj)i they will be 



the successive values of c being in each case arranged at equal 

 intervals in the cycle formed with the indices of the ecjuations 

 (d.) or (e.) taken in order. 



And a similar theorem will exist foryf, T^^) C^O* 



Section IV. 



' 26. The final equation on which j^j, p.^, ;?g depend, is of the 

 (1 .3.4)th degree. This result, which will have been fore- 



