346 



Mk. GOODWIN, ON THE GEOMETRICAL REPRESENTATION 



7. Corresponding to the asymptotes of the curve in the plane of xy there will be infinite 

 branches in space, and it is easy to shew that these go off alternately to positive and negative 



k-TT 



infinity. For from equations (8) we have, when 6 = and p is consequently very large, 



z = p" cos kv = (- ly p" ; 

 therefore for odd values of k the limiting form of the curve is given by 



Z = - p , 



which represents a parabolic branch going off to negative infinity for positive values of p, and 

 vice versa if n is odd, and going off to negative infinity on both sides of the origin if n is even. 

 And for even values of k the form is given by 



^ = p", 



which represents a branch going off to positive infinity for positive values of p, and negative 

 infinity for negative values of jo if w be odd, and to positive infinity in both cases if n is even. 



This proposition it is easily seen includes the real branch of the curve, and hence if we 

 indicate by the mark + or - on an asymptote that the corresponding branch of the curve goes off 

 to positive or negative infinity respectively, the arrangement of the infinite branches will be 

 represented by the accompanying diagram. 



n odd. n even. 



8. I shall next prove the following theorem : 



At points in the real branch of the curve for which the first p differential coefficients of f(x) 

 vanish, there are p imaginary branches going off on each side of the real plane or plane of .vz, 

 and these are curved alternately in opposite senses, the one nearest the real branch being curved 

 in the opposite sense to that real branch. 



Suppose the origin of co-ordinates such that the axis of ss passes through the point in question, 

 which may be done without in any way affecting the generality of the proof, then we shall have 



/(0)=0, /"(0) = /''(0)=0, 



and the equations (7) become 



