498 
Proceedings of Royal Society of Edinburgh. 
images similar. And a peculiarity now presents itself, in that the 
new systems are not directly superposahle : — but each is the per- 
version of the other. 
If we had, from the first, contemplated the question from this 
point of view, an exceedingly simple pair of solutions would have 
been furnished at once by the obviously orthogonal sets of logarithmic 
spirals 
r = a^^ , r = b^~^\ 
and another by their electric images taken from any point whatever. 
The groups of curves thus obtained form a curious series of spirals, 
all but one of each series being a continuous line of finite length 
whose ends circulate in opposite senses round two poles, and having 
therefore one point of inflection. The excepted member of each 
series is of infinite length, having an asymptote in place of the point 
of inflection. This is in accordance with the facts that : — a point of 
inflection can occur in the image only when the circle of curvature 
of the object curve passes through the reflecting centre, and that no 
two circles of curvature of a logarithmic spiral can meet one another. 
We may take the electric images of these, over and over again, 
provided the reflecting centre be taken always on the line joining 
the poles. All such images will be cases satisfying the modified 
form of the problem. 
If we now introduce, as a factor of the right hand member of (1), 
a function of B which is changed into its own reciprocal (without 
change of sign) when 0 increases by tt/2, we may obtain an infinite 
number of additional classes of solutions of the original question ; 
and from these, by taking their electric images as above, we derive 
corresponding solutions of the modified form. We may thus obtain 
an infinite number of classes of solutions where the equations are 
expressible in ordinary algebraic, not transcendental, forms. 
Thus we may take, as a factor in (1), taii2(0 + a). The general 
integral is complicated, so take the very particular case of m=l, 
a = 77 14c. This gives the curves r = a ^ ^ ^ ^^/(i+tanex Again, 
(I -h tan Oy 
let the factor be tan (0 - a) tan {0 + a). With m = 1, and tan a 
= ^/n/ 3, we get the remarkably simple form 
such examples may be multiplied indefinitely. 
^=1 - l ! 
a 
But 
