96 
MR. DAVIES’S GEOMETRICAL INVESTIGATIONS 
interpretation of equation (79.), or of its component ones (80.) and (81.). The 
system is, then, in this case, coincident with that of the curve, whose values of r are 
two, and a circle of infinitely small radius. Still we are not entitled to say that the 
whole system is actually so composed, since by taking the origin of polar coordinates 
in any other way, we should not find the equations expressive of that condition ful- 
filled. It is essential to keep this principle in view (and it is too often overlooked) 
in the discussion of the properties of curve lines. 
In the second place, let us ascertain if (81.) admits of any infinite values of r. In 
this case, the denominator is equal to zero, (since the numerator cannot became infi- 
nite), or, 
(cos a n sin 6 — a ; ) ? — (cos a ; sin 0 — a u )* = 0 (101.) 
This is fulfilled by 
1st, (cos a l sin 0 — — (cos a l sin 6 — a /; )* = 0 1 
and _ t r (102.) 
2nd, (cos ot n sin 0 — a ; )i + (cos a t sin 6 — aj* = 0. J 
Transpose and cube the first of these ; then by expansion and aggregation, we find 
or, 
sin (a ; — a ;( ) cos 6 = 0, 
6 
Hence the radius vector parallel to the magnetic axis is infinite, whilst 6 is finite , 
and hence it is either an asymptote or parallel to an asymptote. To ascertain which, 
let us conceive the origin transferred to V (which, since it is only the position of a 
straight line we are seeking, we are entitled to do) : then, since whether V be between 
the poles or beyond one of them, we have a, — a u = 0 or a l — a tl = r, the equation 
sin (a, — a /( ) cos 6 = 0 is fulfilled by the coefficient of the variable, that line is the asym- 
ptote. This will, however, also appear from other considerations in the next section. 
Again, transpose and cube the second of equations (102.), then we obtain 
tan 6 = = (tan a, + tan aj, 
which indicates the radius vector through the centre M of the magnet, and which, 
since 0 is finite, whilst r is infinite, the line O M is either an asymptote or parallel to 
one. The determination which would be the case, would, from the equations them- 
selves, be not difficult but rather laborious; and hence we shall employ another method 
in the next section to show that it is itself the asymptote to two infinite branches of 
the divergent curve. 
Thirdly. To ascertain whether there be any equal values of r. In this case the 
quantity under the radical symbol becomes equal to zero, or, which is the same thing, 
sec 2 a„— 2 sec a„ sec a, cos 0— cqcosd— a /; + sec 2 a ; ) (cos a, cos a,, sin 0— a ; sin0— 
= sec 2 a // sin 2 0— a ;< (cosa,,sin0— a)$ + sec 2 a ; sin 2 0 — ^(coscqsind— aj* 
