66 Proceedings of the Royal Irish Academy. 



Writing Xi = \xi, we obtain, the equations of the surface 



«i/X = x.,l\f = .r,/,p = X, (29) 



in terms of the parameters A and 6. 



The equation of the tangent plane reduces to 



{\fx, - X.rO ^' + (x, - <j>x,)f'r = 0, 

 which is the polar plane of the point X, \f, (p, 1, with respect to the complex 



0>i2 + xy>34 = 0. 

 Hence the curve given by 



\Yh' = ^ (30) 



belongs to the complex 



Pi2 + cpu = 0, (31) 



and is an asymptotic line of the siu'face. In effect, the polar plane of a point 

 on the curve is the tangent plane to the surface. This contains the tangent 

 to the curve, which is therefore a line of the complex. 



The asymptotic line c is changed into the line c by the transformation 



.r', = X,, z, = x,, x\^c = .Cs v/c, Xis/c = x^^c 



in virtue of (29) and (30). The asymptotic lines are therefore all of the same 

 degree and class, with the same singularities lying on the same generators, 

 exceptions being made for the two dii'ectrices c = 0, c = oo. 

 The foregoing analysis is in substance Pittarelli's. 



Pincli- 'points. 



62. If <p'(6a) = 0, f\0^ ^ 0, two of the generators which can be drawn 

 through the point where the generator ^„ meets S' coincide. This point is a 

 pinch-point. For every finite value of c we find X = for = d,„ and there- 

 fore Xi = X, = 0. Hence all the asymptotic lines pass through the pinch-point. 



Supposing ^ regular when 6 is near ^o, we can write 



^(0) = .^„+(0-e,r-6'. 

 where 6^ is finite for 9 = 0„, unless (^"(0o) = 0. Hence <p' = (6 - 6^) Gi ; and 

 therefore, by (30), for the asymptotic line c, we have 

 X = a(0-0o)^ + J(0-0„)i+ . . . 

 Writing 6 -Bo = t-, we ha\'e \ = at + ht'^+..., ^= <p^ + dt* + hP + . . . 



f =/„ + et- + gt* + . . .; hence xjxt = at + ht^ + . . ., 

 (a>. -/o^OM = X(/-/o) = b't' + ..., (x,- ^o«4)M = ^~f, = dt' + 



Hence all the asymptotic lines pass through every pinch-point on a directrix, and 

 have there an inflexion. They have all the same oseidating plane at tJie point, 

 that, namely, through the pinch-point and the other directrix; aiul the same 

 tangent line, namely, the generator correspondi/ig to the pinch-point. 



