90 
MR. DAVIES’S GEOMETRICAL INVESTIGATIONS 
the axis in the ratio of r , 3 to r„ 3 : and since these radii when infinite are equal, the 
tangent to the infinitely distant point of the curve bisects the magnet, or passes 
through M. 
Now the line M Q 4 being parallel to T P and P' U by construction passes through 
their common intersection; and dividing the axis TU in the ratio of r, 3 to r „ 3 is u 
tangent to the curve at that infinitely distant point. That is, M Q 4 is an asymptote 
to the branch U N 4 of the curve. 
In the same way the other branches are shown to have severally the lines drjiwii 
as already described, through M for rectilinear asymptotes. 
XXV. — The Points of Inflexion. 
In this case the second differential coefficient is equal to zero. 
By (85.) we have 
dy sin 3 0, — sin 3 0„ 
d x sin 2 0, cos 0, + sin 2 0„ cos 0,,’ 
Also 
x + a 
— — = cos 9., or x = r. cos 9, — a , 
r, i i > 
and therefore 
d x = cos 9 , d r { — r { sin 9 1 d 9 t , 
which, since by the triangle 
2 a sin 0, 
sin 0, + 0 ;; 
is convertible into 
r i — 
— (sin 0„ sin 0, sin 0, + 0„ + sin 0„ cos 0, cos 0, + 0„) d 0, 
^ ^ + (cos0„ cos 0, sin 0, + 0„ — sin 0„ cos 0, cos 0, + 0 ;; ) d 0„ 
sin 2 0 ; + 0/; 
Now, by the differential equation of the curve * we also have 
t „ sin 6,d6, . 
d9„ = r-4— -*• 
“ sin 0„ ' 5 
which converts (90.) into 
(90.) 
sin a 0„(sin0,sin0, + 0„ + 005 0,003 0,+ 0//) + sin0,cos0, (cos0„sin0, + 0„ — sin0„cos0, + 0„) 
dx— —2a. r~r — • 9 /i— • d 9, 
sin 0„ siw 0, + 0„ 
or into 
sin 2 0, 
, _ o... cos 0,, + sin 2 0, cos 0 , „ 
dx = - 2a. (l . A . d9. 91. 
sin 0„ sin 2 0, + 0„ 1 v 
In a similar manner from the equation y — r sin 9 t we obtain 
, _ sin 3 0, — sin 3 0„ , . 
dy — —2a . . '. i ‘ d 9. 
3 tin A cin2 A i /) • 
sin 0 /y sin 2 0 y + ''n 
But this is more readily obtained at once from a comparison of (85.) with (91.). 
(92.) 
* Philosophical Transactions, 1835, p. 238. 
f In this case, as also in the formation of the angular equation (85.), d,, is taken the supplement of the 0, 
in the differential equations; and hence the change of sign from — to + . 
