52 Mr. J. Cockle on the Theory of Equations 



tors of innumerable expressions of the fifteenth degree which 

 can be found by combining two and two the values of 0. 



48. Let 



@y.= a-g + ia?, + i^Xc^ + i^x^ + i*x^, 



0/=©',, e^(,.) = ©'„ ©/(,3) = 0'3, 



6/(14) = ®\, ©/(24) =©'5, 0/(34) = e's, 



where, the interchange T^^) being omitted, the relations are 

 expressed by 



0/(«.p)=e'„^^_,, 



and the transpositions T**^} have reference to the roots of the 

 trinomial. 



49. Let ©", ©'", and ©i^ denote what ©' becomes when i is 

 replaced by v^, ?, and i"* successively; then 



©;,+©:+©:'+©': =5*5, 

 ©:.©:©:©;: =^,„, 



for all values of n. 



50. "We also have the alternative theorem*; either 



©; + ©•;=©; + ©{;, (h) 



or 



©;+©!:=©;+©;', w 



where a and b occur in the pairs 



n 21 3-1 



4J' 6J' 5/' 



and (h) holds for the first pair, and (i) for the second and third. 

 51. And, R being a rational function, we have 



where x^, x^ are any two of the four roots x^, x^, Xq, and x^, and 

 a', b' as well as a, b occur in the foregoing pairs. 



* Hence 



and since 



we find that if 



s,=2e',ei"=(e',+e,iv)(e/'+G/") 



=5{a;^5+(a'i+a?4)(a?2 + a;3)}, 



then 3 is the root of a cul)ic whereof the coefficients are rational functions 

 of a?5. 



