PROFESSOR CAYLEY ON POLYZOMAL CURVES. 77 



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satisfying - + T -\ — = 0, or, what is the same thing, taking q an arbitrary para- 



cl 



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meter, and writing - = 1 + q, t- = 1 — g, — = — 2, the equation of the variable 



3. C 



circle is 



ZA° -(- — *— mB° - J ?iC° = . 



1+2 '1-2 2 



Compare Nos. 118-123 for the like mode of construction of a conic; but it is 

 proper to consider this in a somewhat different form. 



181. Assume that the equation of the variable circle is 



D° = (x - dzf + f - d"H 2 = ; 



we have therefore identically 



a A° + bB° + cC° + dD° = 0, 



viz., this gives 



a +b +c = — d, 



a« + bb + cc = — dd, 



and from these equations we obtain a, b, c equal respectively to given multiples 



of d; substituting these values in the equation - + t:"1 — = 0, d divides out, 



and we have an equation involving the parameters of the given circles, and also 

 d, d", the parameters of the variable circle ; viz., an equation determining d", 

 the radius of the variable circle, in terms of d, the co-ordinate of its centre. I 

 consider in particular the case where the given circles are points ; that is, where 

 the given equation is 



Jlk + JwB + JnO = . 



The equations here are 



a +b +c =— d 



&a + bb + cc = — dd 



art 2 + bb 2 + cc 2 = - d(d* - d" 2 ) , 



and from these we obtain 



a ( a - b) (« - c) = - d ((d - b) {d - c) - d" 2 ) 

 b (b -c)(b-a) = - d ((d - c) (d - a) - d" 2 ) 

 c (c - a) (c - b) = - d ((d-a) (d-b)- d" 2 ) , 



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so that the equation - + -r- H — = becomes 



^ a b c 



l(a — b) (a — c) m(b — c)(b — a) n(c — a)(c — b) „ 



(d-b)(d~c)-d" 2 ^ (d-c)(d-a)-d" 2 + (d-a)(d-b)-d" 2 == ' 



VOL. XXV. PART I. U 



