CONCERNING TERRESTRIAL MAGNETISM. 
103 
distance O N describe the circle RNG, which will be the magnetic meridian. 
Through the centres O and O, (O, being the centre of the circle D NC) draw OO,, 
cutting the circles in Q 2 and P 2 : for it will cut them both since it passes through 
both their centres. 
Then (Lemma 8.) the point O, always lies to the left of S, and at a finite distance 
from it. Hence O P 2 is greater than O N, and hence the circle whose centre is O cuts 
that whose centre is 0 ; in another point N. Moreover, the line O O, passing through 
both the centres, is perpendicular to their common chord NN', and bisects both it 
and the arcs N' Q 2 N and N' P 2 N in Q 2 and P 2 respectively. 
But since the line SO ; is finite in all cases (Lemma 8.), the angles N'OP 2 and 
P 2 ()N are also finite, and hence 0 ; K (K being the intersection of N' O with the 
magnetic axis) is also finite, and, obviously, greater than S 0 ( . 
There are, hence, three finite segments of the magnetic meridian in which the nee- 
dle may be placed, distinct from one another, and each requiring a distinct consider- 
ation : and we proceed to prove that in whichever of them placed, except at N, the 
line of its natural direction will not pass through O, the centre of the earth. 
1 . When Q / is taken in the arc of the circle RNG between N and G, as the position 
of a magnetic needle. Then drawing Q y T, Q y U, we have, by Lemma 2., the ratio 
T Q, : Q, U less than the ratio T N : N U ; and hence, drawing the tangent at Q, to 
cut the magnetic axis at S„ the ratio of S, T : S, U, which is the triplicate of this, is 
also less than the ratio S T : S U. The point S ; , therefore, lies to the left of S, or 
more remote from T than S is. Consequently Q ; S, cuts N S on the side of the mag- 
netic axis at H ; , opposite to the centre O of the magnetic meridian ; and as these lines 
have once intersected they cannot intersect again, and hence the line Q, S, cannot pass 
through O ; or, in other words, the needle at Q, in the arc N G cannot be vertical to 
the earth’s surface. 
2. When Q 2 is taken in the arc N' Q 2 N of the magnetic meridian. Draw Q 2 T, Q, U, 
and the tangent at Q, cutting the magnetic axis in S 2 . Then (Lemma 2.) the ratio 
Q 2 T : Q 2 U is greater than the ratio NT : TU; and hence the ratio S 2 T : S 2 U is 
also greater than S T : T U, these being the triplicates of those. The point S 2 falls, 
therefore, nearer to T than S does. The line of the needle’s direction Q 2 S 2 at Q 2 , 
therefore, cuts that at N, at a point H 2 on the side of the magnetic axis opposite to O, 
and hence, as in the last case, cannot pass through O, the centre of the earth. 
3. Neither can a needle placed at any point in the arc RN pass through O. For, 
draw N, S. This is the direction of a needle at N', and hence this does not pass 
through O. Join N' O, cutting the magnetic axis in K. Then S K is a finite quantity, 
and hence the ratio KT : KU is less than ST : S U, and the point A of the curve of 
verticity corresponding to it has its radiants in a less ratio than N' T : N' N. That 
point, therefore, (Lemma 5.) must be more remote from the point of least ratio 
(which, obviously, from Lemma 3., lies on the opposite side of the axis T LI, the angle 
TKO being acute), or beyond the point N. 
