100 
MR, DAVIES’S GEOMETRICAL INVESTIGATIONS 
For every position of ON there is always one single magnetic divergent curve 
which can touch it in the angle U M Y' as at N', each of which will be successively 
found by giving to (3 successive values infinitesimally near to each other. Also as 
these curves approach more and more towards asymptotisin with MY', they approach 
in their course more nearly to the point M : till when MY' becomes the actual po- 
sition of the asymptote, the curve is reduced to the single point M ; and it has been 
shown (XXVI.) that any line, as M O, is a tangent at this point to that individual ease 
of the curve. The curve of verticity which corresponds to the divergent magnetic 
branches in the angle UMY' is, then, finite, and comprised between the lines U O 
and U M within that angle. 
Moreover, as the same line OUQ is a tangent to the infinite branch in the angle 
P M U and to the finite branch in the angle U M Y', these two branches are the one 
continuous of the other. 
Proceeding to the angle Y M T, a precisely similar series of circumstances takes 
place as in its opposite angle U M Y'. The branch has O T for a tangent at T ; it 
proceeds gradually round till it arrives at M and meets the branch UN' M at M. We 
should be led to expect, from the principle of the continuity of the same law holding 
at all points in the course of a locus, that the two branches which meet at M are 
continuous : but as we have no other property of the point M before us except the 
indeterminateness of the tangent to the magnetic curve at that point, we might hesi- 
tate, did any conclusion of importance hinge upon it, to affirm the continuity of those 
branches positively. But by transposing the origin of rectangular coordinates in 
equation (76.) to M, and investigating the number and position of the tangents at the 
origin, the question is settled in the affirmative. The process is, however, long and 
rather intricate ; and as we have no occasion to employ the property in our present 
inquiries, it is unnecessary to give its investigation here. 
In the same manner as in the angle YMU, we may divide its opposite and the 
only remaining region T M Y' into two parts T M P" and P" M Y', and consider them 
in order *. 
The first curve which can have a tangent drawn to it from O is that which has 
OT for its tangent: and as before, the branch thus generated having a common 
tangent with the branch above the axis, they will form a continuous curve at that 
point. 
To all the curves whose asymptotes lie in the angle TM P" there can be one tan- 
gent drawn, and only one : for the point O is on the convex side of the curve viewed 
in reference to the tangent T U at the point of inflexion T of the magnetic curve. 
These will trace out a branch terminating at some point between O and T, as II. 
Whilst the magnetic curves vary through the interval of their passing from R to O, 
the point O will be not only on the convex side of the curve with respect to T U, but 
also between the curve and its asymptote. In this region, then, two tangents can be 
* See also Plate XV. fig. 16, where this part of the work is drawn to a larger scale. 
