of the Fifth Degree. 509 



then these, which I call the /3 and a patterns respectively, differ 

 from the y pattern of art. 62 in this, that 2yS and 2« are not 

 symmetric in 6. But we pass to other 7 patterns by the scheme 



each vertical column of which is such a pattern, Tn this scheme 

 the interchange of 6^ and 6^ marks the transition from the first 

 to the rth column, though in order of derivation from 7, the 

 third column has precedence of the second. 



71. We are thus conducted to a relation of the form 



AA2A3A4A,A6=A'^ .... (k') 



in which all the sinister factors are in the 7 pattern, and which, 

 unless the course of calculation should exclude other patterns, 

 must replace (k). 



72. Next, 7 is of the form 



joQE + gQV + rEo^ + sQx^ + tx^, 



which, introducing 5* for convenience, collapses into 



5''(«QE + &QV + cE^); 

 hence, developing 



27=2 . 5SQE, 2:7,7,= -2 • S'OQ^E^ 

 and reducing, we find 



« + 3c = 2, 



2a2+l2ac + &c + 15c- + 10 = 0, 

 and, by the elimination of a, 



18-3c2 + ie = 0; 

 but when Q = 0, 7= — 2 . 5''Ea^. Consequently 



c= — 2; .•. i = 3, fl = 8, and 



7 = 54(23QE + 3QV-2E^) (n) 



73. I\Ir. Harley has arrived at (n) by a direct process which 

 affords an independent proof that 7 is a rational function of one 

 root only. I ought to state that his calculations (not as yet 

 printed) led me to correct my original values of a, b, and c, and 

 that ho has actually calculated and carefully verified not only the 

 quintic in 7, but also a corresponding quiutic which may be 

 evolved from t,t„ + t.^t^ + T3T5 or t^t^-^ Ir^t^ + t.J.^. 



