and Loci of Apolloiims, ^c. 23 



improvement on a solution by M. Gaultier, in the Journal de 

 I'EcoIe Polytechiiique. 



In tills paper I give ten direct geometrical solutions to 

 the general question. 



The first of these is, I consider, the simplest ever given 

 Its applications to the case in which two of the circles are 

 finite, and the other circle infinitely small, is an improvement 

 on Yieta's solution ; and to the case -where two of the given 

 circles are infinitely small, and the third finite, it is similar to 

 what is given by Brianchon^i^ in the Journal de I'Ecole Poly- 

 technique. 



The second solution is also applicable to all the cases of the 

 problem; and the idea of the auxiliary circle can be applied 

 in other questions, so as to render the solutions more general. 



The application of the third solution to tlie case, in which 

 two of the given circles are finite, and the third infinitely 

 small, leads to 3E. Cauehy's method for this case, &c. 



The other solutions are applicable to all tlie leading cases 

 of the problem, but fail to indicate graphical constructions for 

 some of the minor ones, owing to the peculiarities inherent 

 in the involved theorems, or in the methods of contemplating 

 or expounding them. The tenth is most probably a reproduc- 

 tion of Apollonius' solution. 



The " Loci Problem," which I have undertaken, is in a 

 more general form than was accorded to it by Apollonius. 

 It comprehends almost the entire substance of the Second 

 Book, as restored by Dr. Sirason. 



The solution is direct and general ; besides, it shows that 

 Mhen the ratio is unrestricted in sign, the locus is not (as 

 usually intimated) a circle, but two real circles, a real circle 

 and a point, or a real circle and an imaginary one, according 

 to relative states of the data. 



Particular cases only of this problem were solved by Dr. 

 Simson, all of which have been republished in Leslie^s Geo- 

 metrical Analysis. His methods are inapplicable to the 

 general question, as they depend on the reality of a point in the 

 straight line passing through the given points, which may be- 

 come imaginary, even when the locus is real. 



A method of constructing the locus, having many 

 points approaching to mine, is given in the Geometry of the 

 Library of Useful Knowledge; but there, too, the process 

 depending on points which may be imaginary when the locus 



* Professor Davies has erroneoxisly confounderl Erianchon's %vitli Pappus' 

 solution. — (See vol. ;?, page 227, Mathematician.) 



