50 Mr. J. Cockle on the Theory of Equations 



the symbols of the straight Hues OU, OV, OW meeting the 

 plane X/x.v in A, M, N, and the plane uvw in U, V, W. By 

 substituting A, N for H^ L, and \, f^, v for h, k, I in (p), and 

 then writing M for N, and h, k, 1 for p, q, r in the result thus 

 found, the following equations are obtained: — 



CM + fv + gw OU _ hw. -\-\v-\-\w OV _ \)u + (\v + no OW 

 eX + f/x + gv OA "~ hX + kfi + lv OM ~ p\ + q/i + rv ON " 



Or, writing «', b', c' for the respective denominators of the pre- 

 ceding fractions, 



(cm + XV + gw) — - = [hu + kv 4 Iw) -p- = [\m + q?' + nv) — — . 



Hence by (|) the symbol of the plane uvw, when referred to 

 the straight lines efg, hkl, pqr as axes, will be u'v'iJ, where 



u' = CM + fw + giv, ' 



v' ='}au + 'kv + \iv, I* [v) 



i<;'=pM + qt) + n/;. 



IX. Observations on the Theory of Equations of the Fifth Degree. 

 By James Cockle, M.A,, F.R.A.S., F.C.P.S.'&;c.* 



[Continuedf from vol. xvii. p. 367.] 



42. pUTTING the trinomial under the form 



-*- x^-5Qx^ + ^ = 0, 



the equation in 6{x), or 6, is 



(^ + 55QE6' + 57Qy=5io(108QSE-E'')^. 



43. The assumptions 



fix' = X, 1x^6' = d, 5^Q V = E, 

 conduct us to the still simpler system 



(^3 + 5H^ + 5H)^=(108-H)H2^, 

 in which the accents are suppressed after transformation, and 

 F = 52T3=i(5T)3, H = 5'»TF = 56T''. 



44. In order to apply the formulae of Euler and BezoutJ, I 



* Communicated by the Antlior. 



t Art. 41 is misnumbered "40." In art. 36, line 5, /or " a/5 " read 5. 



X The notation at p. 200 of Ross's translation of Hirsch's Collection is 

 substantially the same as that of Bezout at p. 544 of the Paris Memoires 

 for 1/65, and is more convenient than Euler's at pp. 91, 92 of vol. i.\. of 



