348 Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION 



It would not be difficult from this particular case in which only the first differential coefficient 

 vanishes, to derive the other more general proposition in which the first and any number of 

 subsequent differential coefficients vanish ; at least we could conclude the existence of imaginary 

 branches curved in opposite senses though perhaps not their directions. For we may consider a 



point for which /(v) f"(x) f'i'^) each =0, as the case of p successive maxima and minima 



degenerating into one point, and since these maxima and minima must necessarily occur alternately 

 there will be p imaginary branches curved alternately in opposite senses. 



9. Let us now examine whether the ordinate ^ admits of any maximum or minimum values 

 besides those which it has in the real branch of the curve. 

 The general equation of the curve is 



and the equation for finding the maxima and minima is 



which is equivalent to these two 



/'(O) +/"(O)jOCOS0+/'"(O)^COS20 + +/"(0) '^--- cos(w- 1)6=0, 



11-2 



/"(O) sin 9 +/"(O)|^sin20 + "*■/"(") 1^^ '■"(" - 1)0 = 0; 



and we have also the condition of x being real, which is, 



/(O)sin0+/"(O)£-sin2 + +/"(0)^ sinwfl = 0: 



or these may be written 



Pn-i + '^Pr.-ip cosO + + np'-' cos (n - 1)0 = \ 



2p„_., sin + + WjO»""sin (w - 1)0 = \ (11). 



p„.i sin 6 + p„-ip sin 20 + + /o""' sin w0 = | 



These three equations involving only two unknown quantities cannot be generally satisfied ; I 

 have not been able to shew directly that they never can be satisfied, though it seems possible that 

 such may be the case ; I can however give a complete solution of the question so far as the 

 purpose of this memoir is concerned by proving that a maximum or minimum point is never 

 unaccompanied by a branch curved in the opposite sense, in fact, by extending to all branches of 

 the curve the proposition which has been proved above for the real branch. 



10. The proof is as follows ; 

 We have in general 



where P 



z = f(a; + y y/ - 1) 

 = P + Qv - 1, suppose, 

 d 



Q=(sin,/j/(..). 



