84 
MR. DAVIES’S GEOMETRICAL INVESTIGATIONS 
r 2 cos 2 u lt sin 2 0 — a, {r 2 — 2 b r sec a /; cos 0 — a y/ + b 2 sec 2 a /; } 3 
= r 2 cos 2 a / sin 2 0 — a (( (r 2 — 2 br sec a ; cos 0 — a, -j- 6 2 sec 2 a,} 3 
and this resolves at once into the two equations 
cos 2 a ;l sin 2 0 — a, {r 2 — 2 h r sec a /; cos 0 — eq, -f- b 2 sec 2 a ;/ } 3 4 
( 79 .) 
( 80 .) 
( 81 .) 
= cos 2 a ( sin 2 0 — u tl {r 2 — 2 £ r sec a l cos 0 — a l + b 2 sec 2 a ; } 3 j 
and r 2 = 0 
Jf now we extract the cube root of both sides of (80.), and resolve the quadratic 
for r, we shall obtain the following- equation of the system of branches of the curve 
of contact : 
sec a^cos^ — ^(cosa^sinl — a,) 3 ^ I (scc-a„ 2 sec a^seca^cos^— a, cos£ — -f- sec 2 a,) (cos a, cos sin 6— a^sin^— a,,) 3 1 § 
y — l) — sec«,cos0— agcose^sin 6— a„)^ L — sec 2 a // sin 2 ^ — a ll {cosa ll %in6- '<*■/) * — sec-a/sin 2 ^ — (cos«,sin0— J (82.) 
(cos a„ sin 6 — a,) 3 — (cos a, sin 6 — a„) 3 
From this equation we learn the important fact, that no more than two points of 
the curve of contact can exist for each value of 6. To render it subservient to the 
completion of our object in this inquiry, it will be necessary to establish two other 
properties, viz. that the quantity under the radical is essentially positive, so as to 
render the curve real for all values of 6, and that of these values one is greater and 
the other less than b sec 0. The slightest attention, however, to the form of the ex- 
pression will convince us that this would be a work of great labour if performed in a 
perfectly satisfactory manner ; and that probably it would exceed the means at pre- 
sent in our possession for conducting such a discussion to a successful termination. 
It is fortunately as unnecessary as it is difficult, since by recurring to the genesis of 
the curve itself both these conclusions may be readily established ; and as these are 
all that arc essential to the present investigation, I do not think myself under any 
necessity to examine those characters of the curve which are mere matters of mathe- 
matical curiosity, even though some of them may be very readily obtained from the 
equation itself. Except, therefore, for facilitating some few steps of the succeeding 
course of inquiry, and for the establishment of the above-named general principle (the 
duality of the values of r for each value of 0), I shall rarely have occasion to again 
employ this equation, the objects of its introduction being hereby fully answered. 
XXI. — By referring to art. xix. (pp. 245, 246.) it will be obvious that the geome- 
trical expression of the hypothesis belongs equally to the case of the convergent and 
divergent magnetic curves*; and hence the equations just obtained (81, 82.) must 
express both those cases. Moreover, it embraces the cases where N, O are on the 
* These appropriate terms were first used by Professor Leslie in his “ Geometry of Curve Lines,” p. 400, 
to designate the curves when the poles were respectively of different kinds and of the same kind. Professors 
Robison and Playfair had only considered the convergent curve; and I am not aware that any other author 
except Dr. Roget has taken up the subject. See Library of Useful Knowledge, art. Magnetism, and Journal 
of Royal Institution, February 1831. Mr. Barlow has followed Leslie, Encyclopaedia Metropolitana, p. 794. 
