I. ( 



I i 



It 



vm I 



646 



BEPonr — 1884. 



as containing an iucleforminate angle. Ilenco any two points in the plane niaj' be 

 regarded as subtending any ongle wlmt-soever at .r, and likewise at y. All circles 

 in it therefore pass through these two points on the line at infinity. 



(2.) Let two circles intersect at points A and B not at infinity, and likewise at 

 points .r and »/. The arcs AB in the two circles subtending unequal angles at x, 

 the angle A.cB in the one must he equal to some other A.cC in the other. It 

 readily follows that .r is a point at infinity : t hen, us in (I), that any two lines through 

 it meet at an indeterminate angle, and hence that all circles pass through .r and like- 

 wise through y. 



(3.) Let a circle meet the line <at infinity in .r and y. Its centre C being tlu' 

 pole oi' xy, the radii (Ir and L'// are tangents to tlie circle. Therefore each of them, 

 being also the normal at its point of contact, is at right angles to itself, as conse- 

 quently is every line in the system of parcollels through x or y. 



Join any point O to .c and ?/, draw a circle round O ns centre and let it cut O.r 

 in .r' and Oy m y'. The radii O.r' and Oy' are the normals and therefore also the 

 tangents to the circle at .r' and y', and their chord of contact i^ at infinity because 

 O is the centre. Hence .c' and y are identical with the points .'• and y at intiuity, 

 which are therefore points on every circle in the given plane. 



3. On a Model of the Ciilindroid, shnviuff the Nodal Line. 

 By Professor Robert S. IBall, LL.D., F.li.S. 



On Solvahlr Irrediicible Equations of rrime Degree. 

 By Professor Gkouge Paxtox Youkg. 



§ 1. Let F(,r) = he an irreducible solvable equation of the v«th degr 



I'e, m 



prime, with roots /•,, ?•,,, Sec. The equation b'.ung understood to have been deprived 



of its second term, the roots are of the forms ; 



1 :'■ It 



mr 



mr. 



m-\ 



mr 



0) 



and so on ; where w is a primitive m"' root oi ., . y ; and «,, b^, iX.c., involve only 

 surds that occur in Aj and are thus subordinate to A, . If we call 



A,'", fliAi'", i,A,'", CjA,'" ... (2) 



the separate members of iw\, I propose first of all to establish the fundamental 

 theorem, that the separate mcmhers of the root i\ can be manayed in yrou/is G,, 

 G,,, (Sj-c, such that any symmetrical function of the terms in any one of the yroiqis is a 

 rational f miction of the root (§8). The groups G,, G.j, &c., may be defined more 

 exactly as follows. The ??»*'' powers of the terms in (2) are the roots of a ratioiuil 

 equation of the (?«-l)"' degree auxiliary to l''(,)') = 0. Should the auxiliary not k' 

 irreduci!)le, it can be broken, after the rejection of roots equal to zero, into rational 

 irreducible sub-auxiliaries. This being so, the terms constituting any one of tlu' 

 groups G,, G,_„ &c,, are those separate members of r^, which, severally multi])lii'ii 

 by ni, are ?»"' roots of the roots of the auxiliary, provided the auxiliary be im- 

 ducible ; but when the auxiliary is not irreducible, the terms constituting any one 

 of the groups G,, G„, &c., are »«"' roots of tlie roots of a sub-auxiliHr)'. From the 

 fundamental theorem aliove enunciated can be deduced as a corollary the theorem of 

 Galois, that r, is a rational function (f r., and r ,. In fact, any symmetrical fuuotion 

 of those separate members of r^, which constitute any one of the groups G,, G.^, kc, 

 is a rational function of r., and r^ (§ 1,'}). Not only is it proved that r, is a 

 rational function of /\ and r„, but the inrestiyation shows how the function is farmed. 

 An instance in verification is given (§ 16). It iti^identally appears that if c be the 



