556 Proceedings of the Royal Irish Academy. 



additional propositions that can be given in connexion with these 

 polygons : — 



1. If the alternate vertices 1, 3, 5 ... 2w - 1 of a harmonic poly- 

 gon of 271 sides be joined, the lines of connexion form a harmonic 

 polygon of n sides, and so also do the lines joining the remaining 

 vertices 2, 4, 6 . . . 2n. 



2. If n be an odd number, the three polygons of Ex. 1 are 

 cosymmedians. 



3. If the symmedian lines AK, BK, CK, &e., of a harmonic 

 polygon of an odd number of sides be produced to meet the circum- 

 circle again in the points A', B', C, &c., the points A', B' , C, &c., 

 form the vertices of another harmonic polygon ; and these two poly- 

 gons are cosymmedian, and have the same Brocard angles, Brocard 

 points, Lemoine circles, and cosine circles, &c. 



4. The four symmedian chords of a harmonic octagon form a 

 harmonic pencil. 



5. The circles described through the extremities of the symmedian 

 chords of a harmonic polygon, and intersecting the circumcircle ortho- 

 gonally, are coaxal, and intersect each other at equal angles. 



6. A harmonic polygon of any number of sides can be projected 

 into a regular polygon of the same number of sides, and the projec- 

 tion of the symmedian point of the former will be the circumcentre of 

 the latter. 



7. The symmedian point of any harmonic polygon is the mean 

 centre of the feet of the perpendiculars let fall from it on the sides of 

 the polygon. 



8. If 0,„ 02,1 be the Brocard angles of two harmonic polygons of 

 n sides, and 2n sides, respectively ; then 



tan 0..,, = 4 cos- — . tan 0„. 

 2n 



