CONCERNING TERRESTRIAL MAGNETISM. 
83 
Lemma IX. Problem. — To ascertain which is the greater, T O or T Q. 
Put T O — T Q = c ; that is, c = 
2 a rf r,f 
2 a rf 
( r f ~ r t) i r f + r t r u + r u) i r u ~ r i) ( r u + r,) ( rf + r, r u + rf)' 
Now whilst r H is greater than r p that is whilst the distances T O, T Q are reckoned 
to the left of T, this is essentially positive, since all the factors except r 2 — rf are 
essentially positive, however the quantities be reckoned ; and hence the point O lies 
more remote from T than Q does. In precisely the same way it may be shown to be 
true when the points C and D, &c. are taken respectively to the right of the middle. 
Hence we may infer that, under all circumstances, except those determined in the 
last lemma, the quantity c is finite : and it may be easily shown to increase, as r t and r n 
increase, ad infinitum. 
XX. — On the Points at which the Magnetic Needle takes a Position vertical to the 
Surface of the Earth.* 
At the close of my last paper, art. xix., I stated that I had been unable to resolve 
equation (78.) into its simple or quadratic component factors, and was therefore un- 
able by means of it to assign positively the number of points on the earth’s surface at 
which the needle can take a vertical position. That difficulty may, however, be ob- 
viated by a different process from that which I then indicated ; and as this new 
method fully meets all the objects, physical and geometrical, which led to the for- 
mation of that equation, any further discussion of it in that form may now be dis- 
pensed with. It will here be proved that on the hypothesis of two poles of equal in- 
tensity and of different kinds, there never can he more than two points on the earth's 
surface at which the needle can take the position in question. 
For this purpose let us return to the equation (76.), and take the axis of x parallel 
to the axis of the terrestrial magnet T U (Plate XII. figg. 7, 8.), the centre of the 
earth O being still the origin of the coordinates. Draw O V perpendicular to T U, 
which will coincide in position with the axis of y. Denote the angles which O U 
and O T make with O V by a n and a p the line O V itself in magnitude by b , and the 
current polar coordinates of the curve of contact (of the tangents from O to the 
magnetic curves whose common poles are U and T) by r and 6. 
Then we have V T = a l = b tan u p VU = a n = b tan a lp and b t = b n = b ; and 
likewise x — r sin 6, and y — r cos 0. Make these substitutions in (76.), viz. in 
{b t x— a,y) 2 { (a n — x) 2 + (b u — y) 2 }* = (b tl x - a u y) 2 {(a t — x)* + (b l — y) 2 } 3 ; 
and there will result, after a few reductions too easy and obvious to need indication 
here, the following polar equation of the curve of contact : 
* Continued from the Philosophical Transactions, 1835, p. 248. 
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