298 PROFESSOR DE MORGAN, ON THE 



Encircle any point P by an infinitely small contour, on which let 

 a point be carried round P. Four cases arise ; neither p nor q vanishes 

 within or on this contour ; p vanishes but not q ; q vanishes but not 

 p : or both vanish. 



If neither p nor q vanish, there is never change of sign in either 

 (for by hypothesis they do not become infinite), and the theorem is 

 true for this infinitely small contour : for k and / are both = 0, and 

 there is no radical point. 



If p alone vanish, the curve p = (p being a function of x and y,) 



passes through the small contour at a single or multiple point : and — 



may change sign at those points of the contour through which the curve 

 passes ; but the fraction always becomes and never oo . There are then 

 as many changes of sign from + to — as from — to + , and the theorem 

 is true : for k = I, and there is no radical point. 



If q alone vanish, the curve q = passes through the point : and 



every thing is as in the last, except that - always becomes oo when it 



changes sign. Hence the theorem is true ; for k and / are each = 0, and 

 there is no radical point. 



Lastly, let there be a radical point within, but not on, the contour : 

 which it is evident may be supposed to contain only one radical point. 

 Let x — ft, y = v at the radical point, and let Z be the radius vector 

 drawn from the origin to a point in the contour, and R that drawn from 

 the radical point to the same point of the contour. If then 



m + vj - 1 = A = {a, a); 



we have, using the extended system of algebra, 



Z = A + R, or (»,£) = («, a) + (r, P ), or ase^V- 1 = ae a ^~ l + ri»v^, 



r being infinitely small. Now let <p(A + R), or <pZ, be capable of being 

 expanded into the series 



B R m + B,R m+1 + B 2 R m+ *+ 



in which, on account of the value of R, we need only consider the first 

 term B a R m . 



