TRANSACTIONS OF SECTION A. 607 



G. Note on the General Theory of Anharmonics. Bij A. Buchheim, M.A. 



The paper was based on Clifford's paper on the general theory of anharmonics. 

 It contained a general definition of distances, including Clifford's special definitions 

 and remarks on the extension of the notions of involution and harmonic section to 

 systems of more than one dimension. 



7. Transformations in the Geometry of Circles. By A. Larmoe, B.A. 



There is a well-known theory, due chiefly to Hart, Casey, and Darboux, of th& 

 contact relations of the eight circles which can be drawn to touch three given 

 circles in a plane — viz., that a certain number of groups of four of these tangent 

 circles touch another circle, thus forming two sets of four circles so related that 

 each circle of either set touches all four of the other. 



By treating of plane sections of a sphere instead of circles in a plane, the prin- 

 ciple of polarity is made complete, and the method of inversion, which appears 

 somewhat recondite and artificial in jylatw, is there seen in its true projective light. 

 This generalisation also enables us to deduce the descriptive geometry of a quadric 

 considered with reference to its plane sections. 



The two chief methods of pure geometry that we may use in extending such 

 results when stated for a spherical surface are : — 



(1) If a figure on a spherical surface be connected to any point in space by a 

 cone, this cone will cut the surface again in another figure, which corresponds point 

 for point with the original so that all corresponding angles are equal, and the two 

 figures are therefore similar in their smallest parts though the scale varies from 

 point to point. 



This projection is what, in fact, is known in plane geometry as Inversion. 



(2) 11 we draw the great circles of which the points of the given figure are the 

 poles, their envelope will be the reciprocal figure on the sphere. But this envelope 

 clearly consists of two branches, and the reciprocal character of this transformation 

 leads us to the complete statement of the second principle, which is, that the 

 reciprocal of the original diagram, together with its opposite diagram on the sphere, 

 is the envelope of the polar great circles of all its points. 



Among other consequences the second principle leads to the extension of 

 Casey's results above referred to — viz. if, instead of the three given circles on the 

 sphere, we consider the complete diagram, consisting of the three given circles and 

 their opposite circles, we are led to groups of four of their tangent circles, each of 

 which touches another circle although their members do not touch the same three 

 given circles. 



By supposing the three given circles to become points we deduce, as a par- 

 ticular case of this generalisation, the contact relations of the eight circles which 

 can be drawn through the six points of intersection of three given circles on a 

 sphere or in piano. They are of the same nature as those of the eight tangent 

 circles of three given circles — viz. they can be divided into the same number of 

 groups of four, each tangential to another circle. 



The contact relations of this group do not seem to have been discussed 

 hitherto. 



Casey has also discussed, analytically, the contact relations of the thirty-two 

 conies which can be drawn having double contact with a given conic and touching 

 three conies which have double contact with the given conic, showing that they 

 can be divided into a certain number of groups of four, each of which is tangential 

 to another conic having double contact with the given conic. 



The two principles mentioned above enable us to deduce this proposition \>j pure 

 geometry from the case of the contact relations of the eight circles touching three 

 given circles on a sphere ; and to double the number of groups for which Casey has 

 proved the theorem. 



