MAGNETICAL OBSERVATIONS IN IRELAND. 
151 
The only mode of escaping from these difficulties was to seek 
the general result of the entire series of observations, as to the 
position of the isodynamic and isoclinal lines; and to combine 
the partial results in such a manner that their deviations,—whe¬ 
ther local, temporary, or casual,—should have the least influence 
on the final conclusion. Such is the object of the following com¬ 
putations. 
Let A and p denote the latitude and longitude of any place at 
which an observation has been made, Ay and the latitude and 
longitude of the station which is chosen as the 
origin of the coordinates ; then the position of the 
former place may be fixed with reference to the 
latter in terms of these quantities. For let P be 
the pole of the earth, M and O the two places, 
PM and PO their meridians, and MQ a great 
circle passing through M and perpendicular to 
P O. It is obvious that the position of M will be 
determined by the rectangular spherical coordi¬ 
nates OQ and Q M. Now in the right-angled tri¬ 
angle M P Q, we have 
tan P Q — tan PM. cos P, sin MQ = sin PM. sin P; 
or, denoting the coordinates O Q and Q M by a and /3, 
cot (Ay -r a) — cot A cos (/x, — p t ) 
sin /3 = cos A sin (ju, — 
When g — gy is so small as it is within the limits of the pre¬ 
sent district of observation, we may take 
sin (fc — fc-y) == ^ — p,y, cos (g, — jj* y) - 1, sin y = y, 
and the preceding equatiorfs become 
a — A —Ay, 
/3 ™ (y, — jXy) cos A. v J 
This simplification is obviously equivalent to the substitution of 
the parallel of latitude for the perpendicular to the meridian. 
Now let us conceive any line to pass through O, making the 
angle u with the meridian; then, in the same order of approxi¬ 
mation, the perpendicular from the point M upon that line will be 
p = ft cos u — a. sin u ; 
and, substituting for a and /3 their values just obtained, 
p = (jx — g y ) cos A cos u — (A — Ay) sin u. (C) 
It is easy to see in what manner this result may be applied in 
obtaining equations of condition from the data furnished by 
observation. The increase of the force, or of the dip, may 
(A) 
