( 508 ) 
£ = £«"• . ff-0" = ff-ff', + v «.(5) 
Elimination of n leads, when p and v are neither of them equal 
to zero, to 
By putting 
— *’ *> ?" = «"«*'.. • ( 7 ) 
we find 
.( 8 ) 
The equations (7) and (8) determine together a so-called logarithmic | 
double spiral 1 ), with the points g and g" as poles. 
From the second equation (5) ensues that the angle S' — 6" — tf 
between the two auxiliary radii vectores gz and g"z increases arith- J 
metically uniformly, whilst the first equation (5) shows us that tbe 
quotient of the auxiliary radii vectores increases geometrically uni- | 
formly. 
For the case «, ft y, and <f real, some simplifications appear. 
We shall distinguish three cases. 
A. (a- <f) s + 4*?y> 0, ad- £y < 0. 
The quantities p and q are real, so the points g' and g" lie on the 
real axis. Farthermore we have e y 0, so that v — n. 
Hence the orbit of z is a logarithmic double spiral, whose two 
poles he on the real axis. 
A special case is furnished by the condition a -j- d=0, or g = 0. 
From the first equation (5) now ensues that the quotient of the 
auxiliary radii vectores is constant, so that the point z describes a 
circle of Apollonius of the triangle g'g"z Q , whilst the angle g'zg" in- 1 
creases uniformly with w. An example of the latter case is furnished 
bj ; here Sf = + 1 > / = — 1. 
B . (« — d, 3 + 4£y ]> 0, ad — £y 0. 
The points g' and g" lie on the real axis, whilst e* > 0, thus v = 0. 
Now the second equation (5) shows us, that S' — O" — <p is 
constant, so that the point z describes the circle passing through 
g, g" and z 0 . 
] ) For the logarithmic double spirals the reader may consult: Holzmuller, Ueber 
die logarithm. Abtrildcmg etc Zeitscbr. f. Math. u. Physik., Yol. 16. (1871), p. 281. 
