244 



Mr. R. Lachlan. 



[Mar. 11, 



If (1, 2, 3, 4) denote any system of circles not having a common 

 orthogonal circle, then defining the " power-coordinates " of a point as 

 any multiples of its powers with respect to the system of reference 

 (1, 2, 3, 4), it is deduced from the equation 



=r(!:Ui»=o, 



that the coordinates of any point must satisfy a non-homogeneous 

 linear relation, and a homogeneous quadric relation called the 

 absolute. Also the equation of the first degree in power-coordinates 

 represents a circle, unless it be satisfied by the coordinates of the line 

 at infinity, and then it represents a straight line. 



The equation of the second degree is shown to represent a bi- 

 circular quartic, or a circular cubic, some general properties are 

 proved, and then the curves are classified. It is shown that the 

 equation may be reduced to one of the forms 



ax 2 + ly*+cz 2 + dw 2 =0 (A) 



the absolute being ce 2 + y 2 + z 2 + iu 2 =0 ; 



or arf+lrf+c£=0 (B) 



ax 2 + 2fyz = (C) 



the absolute being x 2 -\-y 2 — 4<zw=0. 



The different curves are then discussed in detail, there being nine 

 species in all, three in each group (A), (B), or (C). 



Part II contains merely the extension of the results of Part I to 

 spherical geometry ; the power of two circles on a sphere is defined to 

 be the product of tan r, tan r\ cos w, where r, /, are the radii, u> their 

 angle of intersection ; the power of a small circle radius r, and a great 

 circle is, however, defined as tan r cos u> ; and of two great circles as 

 cos iv. 



The fundamental theorem is as before 



_ (\, 2, 8, 4, 5\ A 



7, 8. 9, I0j— U > 



connecting the powers of two systems of circles. 



Consequently the results obtained previously are extended with 

 but slight modification. 



In Part III the method of Part I is applied to spheres ; it is proved 

 at once that the powers of any systems of spheres must satisfy the 

 relation 



2, 3, 4. 5, 6\ — 

 ^Kl, 8, 9, 10, 11, 12^ — U ) 



and any of the spheres may be replaced by planes, or the plane at 

 infinity. 



Several results obtained in Part I are easily extended, with one 



