88 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



the point B, or this point is a node. Similarly, the points S and T are each of 

 them a node. 



V. If J I = J in = sin = Jp , 



there are here three branches, each ideally containing (z = 0) the line infinity; 

 the order is thus = 5. Each of the points I, J is an ordinary point on the curve; 

 there are besides at infinity three points, all real, or one real and two imaginary ; 

 that is (besides the asymptotes at 7, J) there are three asymptotes, all real, or 

 one real and two imaginary. Each of the circles J A + %/B = 0, &c, contains 

 the line infinity, and is thus reduced to a line ; the number of nodes is therefore 

 = 3. Hence also, dps. = 3; class = 14; deficiency = 3. 



Cases of the Indecomposable Curve, tlie Centres being in a Line. — Art. No. 196. 



19G. There are some peculiarities in the case where the centres A, B, C, D are 

 on a line ; taking as usual («, £, c, d) for the cr-co-ordinates or distances of the 

 four centres from a fixed point on the line, I enumerate the cases as follows : — 

 I. No relation between I, m, n, p ; corresponds to I. supra. 



II. Jl + Jm + Jn + Jp = 0; corresponds to II. supra. 



III. J I + Jm = 0, Jn + J p — ; corresponds to III. supra. 



IV. Jl + Jm + Jn + Jp = 0, ajl + bjm + cjn + djp = 0; corre- 

 sponds to IV. supra, viz., there is a branch ideally containing (z 2 = 0) the line 

 infinity twice. But, observe that whereas in IV. supra, in order that this might 

 be so, it was necessary to impose on /, m, n,p three conditions giving the definite 

 systems of values J I: Jm ■ Jn : Jp ' — a : b : c ; d, in the present case only two 

 conditions are imposed, so that a single arbitrary parameter is left. 



V. Jl = Jm = Jn = Jp~; corresponds to V. supra. 



VI. Jl + J^, = 0, Jn + Jp= 0, ajl + bJVi + cjn + dJJ- 0, or 

 what is the same thing, Jl : Jm : Jn: Jp = c — d:d — c:b — a : a — b; the 

 equation is thus (c — d)(Jp^ — JW) — {a — b){J^ — Jfrj = . There is 

 here one branch ideally containing (~ 2 = 0) the line infinity twice, and another 

 branch ideally containing (z = 0) the line infinity once ; order is = 5. Each of 

 the points 7, J is an ordinary point on the curve, the remaining points at infinity 

 are a node (A = B°, C° = D°), as presently mentioned, counting as three points, 

 viz., one branch has for its tangent the line infinity, and the other branch 

 has for its tangent a line perpendicular to the axis ; or what is the same thing, 

 there is a hyperbolic branch having an asymptote perpendicular to the axis, and 

 a parabolic branch ultimately perpendicular to the axis. The number of nodes is 

 = 5, viz., there is the node A° = B°, C° = D° just referred to ; and the two pairs 

 of nodes ((c — d) Jfif — (a — b) Jc 5 = 0, — (c — d) JW + (a — b) Jfy = 0) 

 and (c — d) JA° + (a — b) JW = 0, (c — d) JW° + (a — b) J(J = 0), each 



