CONCERNING TERRESTRIAL MAGNETISM. 
91 
To obtain the second differential coefficient, first differentiate the numerator and 
denominator separately. 
d {sin 3 0, — sin 3 0 n } = 3 (sin 2 0 t cos 0^0,— sin 2 0 n cos 0 tl d 0 /y ), "1 
(93.) 
, ,, ,. , . sin 0,d0, t 
or, by the equation d 0 n = c - n g - , r . 
= 3 sin 0 t {sin 0 t cos 0 , + sin 0 lt cos 0 lt } d 0 t J 
d { sin * 1 2 0 l co&0— sin 2 0 U cos 0 H } = (2 sin 0 / cos 2 0 J — sin 3 4 0) d0 t + (2 sin 0 lt cos 2 0 tl — sin 3 0 n ) d 0 lt 1 
= sin 0 t {2 cos 2 6, — sin 2 0 ; ) — (2 cos 2 0 U — sin 2 0 t )} d0 t j> (94.) 
= 3 sin 0 t (cos 2 0, — cos 2 0 /; ) d 0 r 
Hence from (91.), (93.), and (94.), omitting the constant factors, we have 
ein/lrin/lcWjTS (sin 0,0050, + sin0 ;/ cos 0„) (sin 2 0,cos 0,-sin 2 0„cos 0„) - (cos 2 0 ; -cos 2 0 j; )(sin 3 0,- sin 3 0„) 1 
sin 0, sin sin . (sin2 ^ cog ^ + gin2 ^ cog 
sin 0, sin 0„ sin 3 0, + 0„ (sin 2 0„ cos 0, + sin 2 0, cos 0„) 
(sin 2 0, cos 6, + sin 2 0„ cos 0„) 3 
sin 0, sin 0„ sin 3 0, + 0„ (cos 0, + cos 0„) ( 1 — cos 0, cos 0„) n 
(sin 2 0, cos 0, + sin 2 0„ cos 0„) 3 
Since the denominator of this cannot become infinite, the condition is only fulfilled 
by the numerator = 0 : and this gives the five following equations : 
1. sin 0 t = 0 
2. sin 0 tl = 0 
3. sin 3 0,-\-0 ll = 0 
4. cos 0,-J- cos 0 = 0 
and 5. cos0 l cos0 ll = 1. 
(96.) 
The first and second of these show that the poles themselves are true points of 
inflexion, and hence that the order of the branches as to continuity is, that the infi- 
nite are continuous of the finite branches on the opposite sides of the axis, as indi- 
cated by the full line and dotted line representations (Plate XIV. figg. 11, 12.), and 
as stated in (XXII.). It is evident that these conditions are consistent with the 
equation of the curve cos 0 ( + cos 0 U — 2 cos 3 ; and, therefore, all the necessary 
conditions are thus fulfilled. 
The third equation is fulfilled by the equation 0 t + 0 n = x, which is also consistent 
with the equation of the curve. But this is the case when the radiants r, and r n are 
parallel ; and then the tangent, as has been already shown, is an asymptote. The 
view, then, which has been taken at (XXII. o.) of the infinite branches having points 
of inflection on the opposite side of the infinite sphere, is borne out by the analytical 
expression of the points of inflexion. 
The third equation is, moreover, fulfilled also by 0, + 0 H = 0, 0, — - 0 U , which indi- 
n 2 
