203 



Mr. A. J. Ellis on Plane Stiffmatics. 



[June 14, 



focal distances is the axis major in the eUipse or hyperbola. Preserving 

 the same 7iotation, take X' the middle point ofYJ[^, set qfX'X^ = XJL' 

 = 0'S, and make X3Y" = Y2X3, and X^'" From the extremi- 



ties G',H' of the diameter co^^ jugate to that used in the last propo- 

 sitiouy draw straight lines perpendicular to the bisectors of the angles 

 Y"SY2, y'ZYa respectively, forming the two sides of a triangle of which 

 G'H' is the base. Then the lengths of these sides will be mean propor* 

 tionals between the lengths ofSY", SY^ and ZY'", ZY^ respectively. This 

 is the correlative property of the transverse s. foci, which has no Cartesian 

 analogue. The deduction and expression of these properties is extremely 

 simple, and they are illustrated by a geometrical construction. Finally, the 

 s. tangent of the s. angle which the s. rags drawn from any s. point in a s. 

 centric to the two principal or the two transverse s. foci make loith the s. 

 normal at the s. point are eqiial and opposite, vsrhich is a generalization,/o/* 

 both pairs of s. foci, of the property from which the ordinary foci derived 

 their name. 



The following problems are then solved (112). First. Given two con- 

 jugate radii O'E', O'G' to find the principal and transverse axes of the 

 Cartesian centric. The focal stigmata S, Z are found from {e) thus : from 

 E' draw E'M, E'M' perpendicular to O'G', and equal to it in length ; 

 join O'M, O'M', bisect the angle MO'M' by O'S, O'Z, which are in length 

 mean proportionals between O'M, O'M'. Then there are three solutions, 

 giving an ellipse or two hyperbolas, according as both or only one of the 

 conjugate radii meet the characteristic curve of the Cartesian centric. 

 Second. Given any three s. points representing the s. centre, and the s, 

 points of intersection of two conjugate diametrics with the s. centric, to 

 find its principal and transverse axes. Third. Given three s. points, to 

 find the axis of a s. circle passing through them. Fourth. Given any five 

 s. points, to determine whether they lie on a s. centric ; and if so, to find its 

 principal and transverse s. axes. 



Equation (e) is then applied (113) to obtain a m^ore general conception 

 of confocal s. centrics, and Carnot's theory of transversals (as generalized 

 in the Introductory Memoir) is applied to finding a new expression for the 

 s. radius of curvature, applicable for any s. point of contact. The ordinary 

 expression gives length and not direction, and applies only to Cartesian 

 points. Similar investigations are very briefly given for s. acentrics 

 (115-118). 



In these memoirs on Plane Stigmatics, the writer has endeavoured to 

 confine himself strictly to such a development of his theory as would suffice 

 to establish, with geometrical severity, the following results, which he be- 

 lieves to be entirely new, and of great scientific importance to mathematics. 



First. The problem of the geometrical signification of imaginaries in 

 plane geometry is completely solved so that the terms real'' and ima- 

 ginary " are no longer required, and an unbroken agreement exists between 

 plane geometry and ordinary commutative algebra* 



