4< REPORT — i860. 



On some Solutions of the Problem of Tactions of Apollonius of Perga by 

 means of Modern Geometry. By Dr. Brennecke, of Posen. 

 The author suggested a new solution, depending on a remarkable property of the 

 centres of similitude of three given circles; e. y. a circle described around an exter- 

 nal centre of similitude, with a radius equal to the geometrical proportional of its 

 potential distances from the two circles, intersects all homogeneously touching circles 

 orthogonally (around an internal centre all heterogeneously touching circles). Such 

 a circle is called a potential circle. To get the two circles which touch the three 

 given circles simultaneously internally or externally, take two external centres of 

 similitude, draw the two potential circles, find their radical axis, which will contain 

 the centre of similitude of the two circles which cut the three given circles in the 

 same time externally or internally. By combining the three external centres of 

 similitude, you find three potential circles and three radical axes, which all three 

 coincide. Having found this straight line, which contains the centres, it is easy to 

 find the centres themselves by introducing a fourth circle, the reflected mirror-image 

 as it were of any of the three given circles, by means of the found radical axis, and 

 finding out the two circles which touch the two symmetrical circles and anyone of 

 the three given circles. Dr. Brennecke has treated the subject at large in a book 

 which has just now been published at Berlin, ' DieBeruhrungsaufgabe fur Kreis und 

 Kugel,' Th. Chr. Fr. Enslin, I860, Svo, illustrated by eighty-four diagrams, in which 

 all information will be found concerning the most renowned problem of geometry, 

 concerning the problem of tactions of three given circles or four given spheres. 



On a New General Method for establishing the Theory of Conic Sections. 

 By the Rev. James Booth, LL.D., F.R.S. 



On the Relations between Hyperconic Sections and Elliptic Integrals. 

 By the Rev. James Booth, LL.D., F.R.S. 



In this communication the author extended the analogies that the Continental and 

 English geometers had established between elliptic integrals of the third order under 

 the circular form, and the arcs of spherical conic sections, to the corresponding rela- 

 tions between elliptic integrals of the third order and logarithmic form to the arcs of 

 curves described in the surface of a paraboloid. 



On Curves of the Fourth Order having Three Double Points. 

 By A. Cayley, F.R.S. 



The paper is a short notice only of researches which the author is engaged in 

 with reference to curves of the fourth order having three double points. A curve 

 of the kind in question is derived from a conic by the well-known transformation of 

 substituting for the original trilinear coordinates their reciprocals ; and the species of 

 the curve of the fourth order depends on the position of the conic with respect to the 

 fundamental triangle. 



Cn the Triscction of an Angle. By Patrick Cody. 



On the Roots of Substitutions. By the Rev. T. P. Kirkman, A.M., F.R.S. 



To determine the number of roots of a given degree, of a substitution 6 made with 

 n letters, and of the rth order. A substitution 6 which has not two circular factors 

 of the same order, has no roots which are not found among the series 



of its powers. 



A substitution which has two or more circular factors of the same order, will have 

 roots of an order superior to its own, and therefore not among its power. 



Thus the substitution of the 3rd order made with 9 elements, 



. 231564897 



_ 123456789' 



has 1 square root of the 3rd order, 9 square roots of the 6th order, 9 fourth roots 

 of the 6th order, 18 cubic roots of the 9th order, and 18 sixth roots of the 9th I 



