t 431 ) 



the coordinates, lie in the region for which the locus considered is 

 a closed curve. 



So we may sum up what lias been demonstrated as follows. We 



have derived — — = from {<p'") = 0, and stated the condition on 



which a" becomes negative by the substitution of - — = 0. Strictly 



dx 



dtp'" 



speaking we should still have to show that =0 has real roots 



dx 



— and moreover that the value of these roots is in accordance 



with the result obtained. Lot us, for this purpose, examine what 



follows concerning the value of x which satisfies the equation obtained 



before, which we derived trom — — = U, viz.: 



dx 



dA (1— xy— « 9 * a 



= A 



dx x(l — x) [1 — x-\-ir.r\ 



dA c[ ai (l-wy— a^xl] cx(l— x) 



Now — = and A = - and alter reduction 



dx (Y a' 



we find 



or 



a,)Y 4-a„ 

 1 ^ -1 — (l-,tf 



= (1 — xy — n as 



(n-l)« 



For x between and 1 the second member of diis equation has 



a value of n which descends continually and lies between and 



iY(b — £ ) 

 — iv. So there will be a root if L <1 and > - ir. Or if 



{n— ly 



.'«— 1 



n 

 and 



e^e. + Cn-l) 1 

 If* we trace two linos at an angle of 45° with the axes through 



{n—\y 

 the points P and Q, then s x < s, + (n— l) s and ^ ]> e s — 



implies thai has one real root between x = and x = 1 for all 



points between these two lines. It' we confine ourselves to positive 

 values of s, and £ 2 , this space comprises a very large part of the 

 firsl parabola, and moreover the space which 1 shall indicate by OP'Q 



