482 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
(1867) (‘Irish Acad. Trans.,’ vol. 24). In Chapter III. the general theorem is applied to 
several interesting cases of circles; some of the results of this chapter are believed to 
be new. In Chapter IV. the problem of drawing a circle to cut three given circles at 
given angles is considered, and the circles connected with a triangle formed bv three 
circles, which are analogous to the circumcircle, the inscribed and escribed, and the 
nine-points circle of an ordinary triangle are discussed. The results are the same, 
with one or two exceptions which may be new, as arrived at, but in a different 
manner, in the paper by the author in the ‘Quarterly Journal’ (vol. 21). In 
Chapter V. the power-coordinates of a point (or circle) are defined, and the equations 
of circles, &c., discussed ; and it is shown that there are two simple coordinate systems 
of reference; one consisting of four orthogonal circles, mentioned by Clifford (Casey 
and Darboux consider five orthogonal spheres), the other consisting of two orthogonal 
circles and their two points of intersection, which seems to have been indicated for the 
first time by Mr. Homersham Cox in a paper “ On Systems of Circles and Bicircular 
Quartics ” (‘Quarterly Journal,’ vol. 19, 1883). In Chapter YI. the general equa¬ 
tion of the second degree in power-coordinates is discussed, and in Chapter VII. 
Bi-circular Quartics are classified according to the number of principal circles which 
they possess. In Chapter VIII. the connexion between Bi-circular Quartics and their 
focal conics is briefly indicated, the circle of curvature is found, and an expression 
for the radius of curvature at any point of a bi-circular quartic is investigated. In 
these last three chapters the results are probably all old, but as the method employed 
is different from any previously used to discuss these curves in detail, it may not be 
without interest. 
In Part II. (§§ 125-198) almost all the results given in Part I. are extended, with 
occasionally some slight modifications, to the case of small circles on a sphere and 
spheri-quadrics. 
In Part III. (§§ 199-287) the same order is followed as in Part I.; most of the 
results in Chapter III., Part I., are extended to the analogous systems of spheres. 
In Chapter III., however, it is shown that though there is a group of spheres 
corresponding to the circum-sphere of a tetrahedron, and though several analogous 
theorems are true for what correspond to the inscribed and escribed spheres, yet there 
is no analogy to Feuerbach’s theorem. Chapter IY. corresponds exactly to 
Chapter V. in Part I., and in Chapter V. the general equation of the second degree in 
power-coordinates is shown to represent a cyelide, and the equation is discussed in the 
same manner as in Part I., Chapter YI. The reduction, however, of the general 
equation to its simplest form presents some difficulty. In Chapter YI. cyclides are 
briefly classified in the order in which they present themselves in reducing the general 
equation, and in Chapter VII. a few miscellaneous propositions are discussed, as, for 
instance, the determination of the locus of the centres of the bitangent spheres, i.e., 
the Focal Quadrics. 
It may be convenient to state here the Memoirs consulted :— 
