296 Proceedings of Boyal Society of Edinburgh. [sess. 
a 10 = r 10 • cos (60) = (x 2 + y 2 ) 2 • r 6 • cos 66, 
= (; x 2 + y 2 ) 2 {x 6 - 1 5# 4 y 2 + 1 5x 2 y^ - y 6 }; (3) 
a™ = cos (i6) = (x 2 + y 2 ) 6 [x^ - 6x 2 y 2 + y^} ; .... (4) 
a 10 _ r io cos ^2 6) — (x 2 + y 2 Y{x 2 - y 2 } (5) 
To these may be added — 
a io _ r io . cos 2^ . cos 5 6 ; a 10 = r 10 cos 20 • cos 4 6 ; a 10 = r 10 cos 26 • 
cos 36; a 10 = r 10 cos s (26) ; 10 10 = r 10 cos 3 (30). 
When these ^-and-y equations are fully expanded, they are found 
to he central homogeneous equations of the 10th degree, each having 
its distinct combination of + and - signs. Numbers (1) to (5) are 
all hyperbolic curves without inflexions, and having the branches of 
each curve all equal and all symmetrically disposed about a centre. 
These characteristics evidently apply to all curves of even integral 
degrees so obtained. 
In (1) there are 10 asymptotes, with angular intervals of 18°, and 
10 real branches. 
In (2) there are 8 asymptotes, with angular intervals of 22J°, and 
8 real branches. 
In (3), (4), and (5) the number of asymptotes and branches are 
respectively 6, 4, and 2, and the angular intervals are 30°, 45°, and 
90°, the circle being always equally divided. The curves of the 
equations containing two cosines have not been examined. The 
above are all the symmetrical homogeneous hyperbolic curves that 
can be formed in the 10th degree, and for any degree the number 
of such curves is evidently w/2. They are all curves of “ perfect 
symmetry” in the sense defined in the introductory paragraph. 
If m exceed n , as, for example, in the expression a 10 = r 10 cos (120), 
we obtain by transformation to rectangular coordinates a homogeneous 
equation of the second class (so termed in my paper on this subject), 
i.e., of the form <}> n (x, y) = 4> p (x, y), or = 1. The number of 
VA 00 j V ) 
such curves (all traceable) for any degree, n —p, is infinite. 
For the equations of the form r n = a n • cos (m6) or a n = r n sec (m6), 
above considered , 
we also obtain homogeneous Cartesian equations of the second 
