284 Proceedings of Royal Society of Edinburgh. 
Writing w for i, and putting for p its known value, p = r and 
for x and y their values, r cos 0 and r sin 6 — the equations (3) 
become — 
■v ^ 1 dr — 
A = r cos v - ^ r • • cos w • cos 2o> ; 
2 dp 3 
-a 1 dr 
Y = r sin 6 - — r • — - • cos to • sm 2cu. 
2 dp 
From these equations it is evident that if p and w were known in 
terms of r and 0, one of these pairs of variables might be replaced by 
the other, and the equations integrated. This condition is fulfilled in 
the case of all curves which can be expressed in the two-term polar form 
a n = r n • cos (m6) 
j a" = r" • cos {mV) 1 
| a n = r n • sec (in6) j 
where m and n are any numbers, integral or 
I IIW I J 
fractional. 
It is proposed to investigate the caustics of curves expressed in 
this form, and thereafter to determine their equivalents in £-and -y 
coordinates. It will be shown that the equation of the caustic by 
reflexion is of the form — 
E = a • sec* (pO) • sin ( q6 ) ; 
or, E = a • cos* (pO) • sin (qO) . 
where Z, p, q , are given in terms of m and n. 
2 . 
Systems of Derivative Curves. 
In a system of curves of different degrees, each of which is the 
pedal of the curve preceding it, and the negative-pedal of the curve 
following it in the series, by a known theorem, all the tangents of 
corresponding points of the system make equal angles with their 
respective radius- vectors. If S m , S m _ 1? &c. (fig. 1), be correspond- 
ing points in such a series of curves, all the angles OS TO S m _ 1} 
S m _ 2 , &c., are equal. As the adjacent angles are right angles, 
it follows that the triangles are similar, and that all the angles at 
the centre, v m , v m -i> &c., are equal. In other words, if v m be the 
angle between the radius-vector and the perpendicular on the tangent 
for any curve ; v m+1 the corresponding angle for the negative-pedal, 
and v m _! the corresponding angle for the pedal — then 
i+i 
( 5 ) 
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