358 



Mb. GOODWIN, ON THE GEOMETRICAL REPRESENTATION 



But if — be < — , the form of the 



8 27 



curve will be essentially different, and 

 will be as in the annexed figure. If we 

 suppose the figure to correspond to the 

 case of r positive, then the figure for r 

 negative will be found as before by sup- 

 posing everything turned through two right 

 angles about the axis of %. 



.(31); 



23. The curve corresponding to the equation ai" - 1 = is easily traced, and furnishes a good 

 illustration of what precedes. I shall trace this curve with polar co-ordinates. 



We have, ar = p" (cos w0 + \/- 1 sin w0) - 1 (30), 



whicli divides itself into the two equations, 

 z = p" cos nd - 11 



0= sin w0 J 



from the latter of these nd = kir where k = 0, 1, 2 (n - 1) ; 



.-. a- = (- l)y _ 1. 



Hence the complete curve will consist of a series of parabolic curves defined by the equations 

 •7 = p" - 1 and as = - p'' - \ alternately, and lying in planes 

 passing through the axis of z and making with each other an 



angle — . 



The figure represents the curve; Oa„ Oa.^ are the 



branches stretching up to positive infinity, Oa„, Oa^ 



those to negative infinity : the plane of xy intersects the 

 former set of branches but not the latter, and gives for the 

 roots AX.,, AX.^ 



If we suppose the plane of xy to intersect the branches 

 Oa^, Oa, we should have the case of the equation 



i»" + 1 = 0; 



and if the plane were to pass through O, we should have the 

 curve corresponding to x' = 0, in which case the roots would 

 be all equal to 0. 



