1894-5.] Prof. Tait on Systems of Plane Curves. 
497 
Systems of Plane Curves whose Orthogonals form a 
Similar System. By Prof. Tait. 
{Ahstract.) 
(Read May 6, 1895.) 
While tracing the lines of motion and the meridian sections of 
their orthogonal surfaces for an infinite mass of perfect fluid dis- 
turbed by a moving sphere : — the question occurred to me “ When 
are such systems similar ? ” In the problem alluded to, the equa- 
tions of the curves are, respectively, 
(r/«)2 = cos0, and (r/by = sin d . 
It was at once obvious that any sets of curves such as 
(r/ a)'^ = cos 0 and {rlh)m = sin 6 
are orthogonals. But they form similar systems only when 
= 1 . 
Hence the only sets of similar orthogonal curves, having equa- 
tions of the above form, are {a) groups of parallel lines and (&) their 
electric images (circles touching each other at one point). As the 
electric images of these, taken from what point we please, simply 
reproduce the same system, I fancied at first that the solution must 
be unique : — and that it would furnish an even more remarkable 
example of limitation than does the problem of dividing space into 
infinitesimal cubes. (See Proe. vol. xix. p. 193.) But I found 
that I could not prove this proposition ; and I soon fell in with an 
infinite class of orthogonals having the required property. These 
are all of the type 
>-^ = (tan6») (1). 
which includes the straight lines and circles already specified. The 
next to these in order of simplicity among this class is 
1 
r = ae 2 cos20 cos 0 ., 
^ 
with r = b&^ sin 0 . 
In order to get other solutions from any one pair like this, w’e must 
take its electric image from a point whose vector is inclined at tt/I 
or Stt/I to the line of reference. For such points alone make the 
VOL. XX. 2 I 
