TRANSACTIONS OF THE SECTIONS. 15 



of the diagonal of a square, as shown in the first proposition ; (2) a property of this 

 triangle is that each of its equal angles is equal to 1^ the imequal angle +i the 

 angle included by the straight lines joining the unequal angle to the points of tri- 

 section of the unequal side ; (3) another property is that the unequal angle can be 

 divided into two parts such that the square on the chord of one segment is double 

 the square on the chord of the other segment (a similar property belongs to the 

 isosceles right-angled triangle) ; (4) the square can be reduced to what may be 

 termed its elementary triangles, eight in number, all the angles having a definite 

 relation. 



The paper treats of the properties of this triangle in combination with the circle, 

 the square, circles in geometrical progression, two and three circles, the square and 

 the circumscribing circle, and the ellipse. 



Two Memoirs. — I. On the Shadows of Plane Curves on Spheres. II. On 

 Oubio Spherical Curves with triple Cyclio Arcs and triple Foci. By 

 Henbt M. Jeffekt, M.A. 



I. On the Shadows of Plane Curves on Spheres *. 



1. M. Chasles, in his Geometrical Memoirs on Spherical Conies (which laid the 

 foundation of the subject), has investigated several of their properties from projec- 

 tions of the circles lying in a cyclic plane of the cone. 



It was proposed to establish general analytical processes which should embrace 

 these theorems, particularly as that geometer has urged the subject on analysts. 



2. The several systems of coordinates in ordinary use were adapted from plane 

 to spherical geometry. 



Cartesian coordinates are reduced from gnomonic projection to Gudermann's 

 system, in which the coordinates of a point, whether rectangular or oblique, are 

 tangents of the arcs intercepted on the arcs of reference. From gnomonic projec- 

 tion, Boothian tangential coordinates are represented on the sphere by cotangents 

 of the arcs intercepted on the arcs of reference. 



Ex. The focal equation to the plane conic 



- = l-|-ecos^. 



The equation to its projection on the sphere has the same form^ 



tan Z T 8iii2y . 

 tan p sin 2a 



for 



_ A'S— AS ^ tan 8'— tana ^ sin (S'-8) ^ sin2y 



* — A'S + AS tan 8'+tan S sin (8'-)-8) sin 2a ' 



where the symbols have the ordinary acceptation. 



The same process was shown to be applicable to determine the analytical forms 

 and geometrical properties of both pole- and polar-spherical curves. 



3. Equations to a circle or conic, which are expressed in rectangular coordinates 

 in a cyclic plane of a cone, were converted into three-point tangential equations of 

 the projected spherical ciu-ve. 



Two points of reference are situated in the cyclic arc, and the third is the polar 

 point of the cyclic arc with respect to the spherical conic. By this process the several 

 properties of the spherical conic which relate to a single cyclic arc are simply 

 deduced from those of the plane circle, in following the geometrical guidance of 

 M. Chasles. 



4. Formula3 were next given to express the shadow of a plane curve, as deter- 

 mined by tiilinear coordinates, by spherical coordinates. 



* This memoir has been printed in extenso in the ' Quarterly Mathematical Journal,' 

 1875. 



