347 
520] ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
if be the value at the outcrop. Writing 8s for the element of the arc we have 
cilV:3 
which exhibit the form at the outcrop. 
The Nodal Curve; expressions for the coordinates in terms of a single parameter a. 
Art. Nos. 53 to 60. 
53. After the foregoing investigation of the nodal curve, I was led to perceive 
that it is possible to express £, r/, £ 1 , rjj in terms of a single variable a, and thus to 
obtain expressions for the coordinates of a point of the nodal curve in terms of the 
single variable a. The result was obtained by the consideration that the acnodal 
portion of the nodal curve could only arise from imaginary values of £, tj ; the question 
thus was, what imaginary values of these quantities give real values for the coordinates 
x, y, z. To make y real we may assume 
% = -b--p (6 - (pi), 
V = — If- + p {6 -F (pi) 3 , 
(i = \J — 1 as usual): this being so, if A denote one or other of the quantities 
7, — a (= a 2 — h 2 , c 2 — h 2 ), 
the expressions for — (3ya 2 x 2 , — 7ab 2 y 2 will be 
= (A— p{6 — <pi)} 3 {A +p(6+ (pi) 3 }, 
and we have therefore the condition that this shall be real (for the two values A = 7, 
A = — a) : being real, it will in certain cases be positive, and we shall then have real 
values for the remaining coordinates x, z. 
54. The condition of reality is easily found to be 
A 2 (3(9 2 - <p 2 + 3) - 60pA (6> 2 + </> 2 + 1) + p 2 {3 (6 3 + (pj + 36 2 - </> 2 } = 0, 
viz. this equation in A must have the roots 7, - a, or the expression on the left hand 
must be 
= (30 2 — <p- + 3) {A 2 — (7 — a) A — 0C7}: 
we have therefore 
(7 — a) 2 36(d 2 +^> 2 + l) 2 
— 7a (30 2 — <p 2 + 3) {3 (6 2 + cp 2 ) 2 + 3d 2 — <p 2 } ’ 
6 dp (6- + 4> 2 +1) _ 
7-a ^ 30 s — <p 2 + 3 ’ 
44—2