90 MR. BROOKE ON THE AUTOMATIC REGISTRATION OF MAGNETOMETERS, 
These equations, solved by the method of least squares, give 
^=13-0269; 3/ = 0-0685. 
By the method of weights described above, the scale value of O'OOl of the total hori- 
zontal force was found to be 25*27 sc. div., consequently the values of x and y ex- 
pressed in parts of force are 
:c=0-00051550; ?/=0000002710. 
As the elfect of a small increase of weight in displacing the magnet is the same as 
that of a small corresponding diminution of force, when the correction is subsequently 
applied to the magnet suspended in air, the value of x above found must be dimi- 
nished by a small quantity representing the effect of the increase of weight due to 
the weight of the water displaced by the magnet being diminished by expansion, as 
the temperature increases ; this may be called the coefficient of expansion, and may 
be thus found. 
Let w, s, b be the cubical expansions of water, steel and brass respectively, for 1° 
Fahr., then 
^4;=0■0001500 (Ure) 
s =0*0000204 (Smeaton) 
6=0*000029] ; 
and if U be the weight lost by the magnet at 32° Fahr,, and U' the weight lost at any 
observed temperature, 32°+^°, the value of U will be 
U'{1+(m;—^)^} = U'(1 +0*0001296 1 ) ; 
and if V and V' similarly represent the weights lost by the stirrup and end of the frame 
immersed, 
V=V'{1 + (m; — 6)^}=V'(1+0*0001209 t). 
Let W be the weight in air of the magnet and its appendages, then the coeflScient of 
expansion for f above 32° expressed in parts of the force will be 
UxO-0001296 + Vx 0*0001209 
W 
In the present instance the value of this coefficient is 
0*00001022 t, or 0*0000 1^ nearly, 
and the value of x being diminished by this quantity, the whole temperature coefficient 
will be 
0*00050528 ^+0*000001 355^2. 
The temperature coefficient of the bar (C.B. IX.) similarly determined is 
0*0003822 ^+0*000000947^2. 
In both these instances, it will be seen that the temperature coefficient varies con- 
siderably from the uniform law usually employed in the reduction of magnetic ob- 
