are conjugate points with respect to the conic ax2+By2+ yz2=0, is 1 1 1 + + = 0. α B γ Or writing this equation under the form By+ya+aß=0, and substituting for a, B, y their values, we have the equation already found, as the condition in order that it may be possible in the conic a2+ y2+z2=0 to inscribe an infinity of triangles the sides of which touch ax2+by2 + cz2=0. Theorem. Let ax2+by2+cz2=0 be the equation of a spherical conic, and let (: : ), a point on the conic, be the pole of a great circle cutting the conic in two points; the conic intersects upon the great circle an arc given by the equation cos &= (a+b+c) √ E2+n2 +52 √ (a+b+c)2 (§2+n2 +52) −4(bc§2 + can2+ab52) Hence if a+b+c=0, 8=90°; or there may be inscribed in the conic an infinity of triangles having each of their sides equal to 90°. It is worth while, in connexion with the subject, and for the sake of a remark to which they give rise, to reproduce in a short compass some results long ago obtained by Jacobi and Richelot. The following are Jacobi's formula for the chords of a circle, subjected to the condition of touching another circle; viz. if in the figure we put MA2=cA2-cM2=a2+ R2+2aR cos 24-72 =(a+R)—2—4aR sin2 has the same value as before. Hence the rela tion between 0, 0' is de ᏧᎾ ✓1-k2 sin20 √1 — k2 sin2 013 which is identical with that between 4 and p'; and in fact the equation between 0, & is which, if amu, gives 0=am(K—u). The differential equation contains only a single arbitrary parameter; hence the same differential equation might have been obtained from different values of a, R, the parameters which determine the circle enveloped by the moveable chord. The m' m condition for this of course is = i. e. n i. e. (aa'-R2) (a-a')-R(a'r2—ar12)=0, which implies that the enveloped circles intersect the other circle in the same two points, or that all the circles have a common chord. Let the corresponding value of e' be 'a, i. e. suppose that for 0=0, we have '=a, then And a having this value, we have for the finite relation between 0,0', FO-FO+Fa. = Richelot has shown, by precisely similar reasoning, that for circles of the sphere we have dif = cos2r-(cos R cosa + sin Rsina cos24) which is of the form And it is very important to remark, that this equation contains the two parameters A, μ, so that the same equation cannot be obtained with any new values of the parameters a, r; or the formulæ in plano for three or more circles do not apply to circles of the sphere: the geometrical reason for this is as follows, viz. in the plane a circle is a conic passing through two fixed points (the circular points at co), and consequently any number of circles having a common chord are in fact to be considered as conics, each of which passes through the same four points. But circles of the sphere are not spherical conics passing through two fixed points, but are merely spherical conics having a double contact with an imaginary spherical conic (viz. the curve of intersection of the sphere with a sphere radius zero); hence circles of the sphere having a common spherical chord are not spherical conics passing through the same four points. I am not sure. whether this remark as to the ground of the distinction between the theory of circles in plano and that of circles on the sphere has been explicitly made in any of the treatises on spherical geometry. To reduce the equation, write Let the corresponding value of 0' be '=a, i. e. suppose that for 0=0, we have '=a, then And a having this value, the finite relation between 0, 0' is FO=F0+ Fa. By comparing with the corresponding formula in plano, wc arrive at Richelot's conclusion, that the formula for the sphere may be deduced from those in plano by writing in the place of Ra tan R sin a the functions ܕ r 2 Stone Buildings, October 1, 1856. tanr' cos R sin respectively. r |