102 
MR. DAVIES’S GEOMETRICAL INVESTIGATIONS 
XXX. — Professor Leslie’s Property of the Magnetic Curve, and a Genesis of the Curve 
of Verticity founded on it. 
If tangents be drawn from a given point in the magnetic axis to a series of mag- 
netic curves, either convergent or divergent, the locus of the points of contact is a 
given circle. 
For since the point S is given (Plate XVI. fig. 19.), the ratio ST: SU is given, and 
hence its subtriplicate NT:NU is also given; and the points T and U being also 
given, the conclusion follows from Lemma I. 
The same conclusion also follows from our equation (76.), as in this case b l =b ll = O 
(taking the axis of a parallel to TU), and the equation is converted at once into 
y af { (a (/ 
or if — 0, 
— jo ) 2 +y 2 } - y «/ { («/ 
and [af — a f) (x 2 -\ -y 1 ) — 
x y _|_ yi\ s — 0j 
■aj a u‘ { af — a/} x-\-af af {af — af} 
(104.) 
The former of these is a foreign factor introduced, so far as the locus of a lower 
order than 76 is concerned, by the eliminations through which that equation was ob- 
tained : the latter is the equation of a circle whose centre is in the axis, but not in the 
form best adapted for use; which would be to refer it to M, and thereby make 
— a u — a i = a. As, however, we only require it for constructive purposes and geo- 
metrical reasoning, it is unnecessary to examine the equation further ; and, except as 
a verification by a particular case of our general equation (76.), would not have been 
noticed here. Where it is possible, such verifications, it is admitted on all hands, 
should be made. 
Genesis of the curve of verticity. — Take any point S in the magnetic axis, and find 
two lines in the triplicate ratio of ST:TU*. Describe the circle DNC, which is 
the locus of lines inflected from T and U in this triplicate ratio, (Lemma I.,) and let 
it cut the line O S in N. N is a point in the curve. 
This does not enable us, however, to discriminate the branches themselves of the 
two classes of curves ; nor, therefore, supersede the necessity of the preceding inves- 
tigations. 
XXXI. — The Circle whose centre is O cannot cut the asymptotic branches of the con- 
vergent system of the Curve of Verticity in more than two points, one in each branch. 
Let N be a point in the curve of verticity where it cuts the earth’s surface such that 
O N is the direction of the needle tending the centre of the earth. Draw the radiants 
N T and N U ; and let N O intersect the magnetic axis in S. Then ST : S U is the 
triplicate of the ratio N T : N U, or of r ; : r ir Describe the circle DNC (Lemma 1.), 
which is the locus of the point N corresponding to S ; and with centre O and 
* I)r. Roget, Secretary of the Royal Society, has given a very elegant construction of this problem in the 
Journal of the Royal Institution for February 1831. 
