344 



Me. GOODWIN, ON THE GEOMETRICAL REPRESENTATION 



= (cost/— - + \/^ sin y-—) /(«), 



which equation divides itself into these two 



.(6). 



If the symbols in these expressions be expanded it is evident that equations (3) and (6) will 

 coincide. 



There is another mode of expressing the equations in question which will be found very 

 useful in the sequel, and that is by polar co-ordinates. 



Put X = p cos 9, y - p sin 9, then 

 X =f(p cos 9 + \/~ I p sin 9), 

 which divides itself into t\vo 



^=/(0) +f\0)pcos9+-^~^^^ p'coso0+ ... +-Q^p"coin9 



ll 



[- 



= 



/'(O) sin 9 + -Q-^ p sin 29+ ... + ^-^ p"-' sin ni 



.(7); 



If 



\n 



(iiny ~\f(x.a:) = 



■(B), 



the dift'etentiation indicated being partial with respect to a-. 

 Ot' course we might have treated z in equation (^-1; in the sume 

 manner as .r, and this would have given the following 



/(..■■ 



) = o 



(sin3,^^)A.,..^) = 



■((')■ 



The equations (B) and {O may he considered as the complete 

 representation of the locus of ( ^ ). 

 For example, suppose 



/(j-.«)= -5+rj-l, 



then 



df(x.x) _ 2x df(i-z) ^ 25 

 dx ' a^ dz ~ /,' ' 



d' f(x.z ) 2 d'f{x.z )_2 

 dx' a- dz' i'' 



and equations (S) become 



and equations ( C ) become 



.ry =0' 



■ (B-) 



■ (.C), 



and it will be seen that the systems (S') ( C) are equivalent to 

 these three, 



7/ =0 J .1=0' i? =0 ' 



Or the locus of the ordinary equation of the ellipse, thus 

 considered, comprehends an ellipse and two hyperbolas, the two 

 hyperbolas setting oft'in planes perpendicular to that of the ellipse 

 from the extremities of its axes. 



I would refer here to two papers in the Cambridge Mathema- 

 ticaljouriialjhy Mr. M''alion, of Trinity College, (Vol. 11. p. 103 

 and p. 155) in the first of which the comple'e representation of the 

 curve corresponding to a given equation between two variables is 

 considered, and in the second the real nature of a maximum or 

 minimum as being in fact a multiple point is noticed. 



