646 BEPonr— 1884. 



as containing an indeterminate angle. Hence any two points in the plane may be 

 regarded as subtending any angle whatsoever at x, 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 x and y. The arcs AB in the two circles subtending unequal angles at x, 

 the angle A.rB in the one must be equal to some other A.rC in the other. It 

 readily follows that x is a point at infinity ; then, as in (1), that any two lines through 

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

 wise through y. 



(3.) Let a circle meet the line at infinitj- in x and y. Its centre C being the 

 pole of xy, the radii Cx and Cy are tangents to the 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 parallels through x or y. 



Join any point O to x and y, draw a circle round as centre and let it cut Ox 

 in x' and Oy in y'. The radii Ox' and Oy' are the normals and therefore also the 

 tangents to the circle at x' and if, and their chord of contact is at infinity because 

 is the centre. Hence x' and y are identical with the points x and y at infinity, 

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



3. On a Model of the Cylindroid, showing the Nodal Line. 

 By Professor Robert S. Ball, LL.D., F.R.S. 



0) 



4. On Solvable Irreducible Equations of Prime Degree. 

 By Professor George Paxton Young. 



§ 1. Let F(.r) = be an irreducible solvable equation of the with degree, m 

 prime, with roots r v r v &c. The equation boing understood to have been deprived 

 of its second term, the roots are of the forms : — 



L 2. 3 m-l 



mi\ = Aj M + Oi A,'" + 6^! + . . . + Cj Aj '" 



1 2 5 m-1 nM 



mr., = coA! 1 " + <B 2 fl, A,™ + oj^A/'" + . . . + co c l ^ l '" 



12 3 (»1-1) m-1 



m?- 3 = co-A 1 m +a> 4 a 1 A 1 '" + ui l ''b i A 1 m + ... + w : c^V , 



and so on ; where w is a primitive m ih root of unity ; and a v b u &c, involve only 



i 

 surds that occur in A : and are thus subordinate to A x . If we call 



1^ 2_ 3^ m-l 



Ai'", fljA/", b.A,'", CjAj"' ... (2) 



the separate members of mr lf I propose first of all to establish the fundamental 

 theorem, that the separate members of the root r, can be managed in groups G lt 

 G 2 , tyc, such that any symmetrical function of the terms in any one of the groups is a 

 rational function of the root (§8). The groups G v G 2 , &c, may be defined more 

 exactly as follows. The «t th powers of the terms in (2) are the roots of a rational 

 equation of the (?H-l) th degree auxiliary to F(.r) =0. Should the auxiliary not be 

 irreducible, 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 the 

 groups G„ G 2 , &c, are those separate members of r lt which, severally multiplied 

 by m, are m th roots of the roots of the auxiliary, provided the auxiliary be irre- 

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

 of the groups G„ G 2 , &c, are »j th roots of the roots of a sub-auxiliary. From the 

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

 Galois, that t 1 is a rational function ofr 2 and r r In fact, any symmetrical function 

 of those separate members of r u which constitute any one of the groups G 1( G.,, &c, 

 is a rational function of r s and r 3 (§ 13). Not only is it proved that i\ is a 

 rational function of r„ and r.,, but the investigation shows how the function is formed. 

 An instance in verification is given (§ 15). It incidentally appears that if c be the 



