Terrestrial Magnetism. 167 
da bie or Bopabihb ph 200 bt 
: ~ pa cos (8—8) ’ ¢  posin(o—8) ’ 
or, denoting the limit of error arising from friction in the ordinary position of the 
needle by « , 
d3 = ¢ sec (3-6) , tae cosec (3—0). (7) 
We learn, then, that the error in the determination of the dip, arising from friction, 
is least when 8—0=0; and that the smallest value of d8 is ¢.* The corresponding 
error in the determination of the force will be a minimum, when §—6@=90° ; in which 
d 3 A ‘ 
case ee fee being expressed in parts of radius. 
As far then, as friction is concerned, it would appear to be the most advantageous 
modification of the method suggested in the preceding pages, to observe the position 
of the needle, in the first instance, without any counterpoise, and, afterwards, with a 
counterpoise which will bring it into a position nearly perpendicular to the line of the 
dip. ‘The former of these angles is the dip itself; and the two angles, when substi- 
tuted in formula (4), furnish the measure of the intensity. But, in order to avoid 
the error arising from the non-coincidence of the centre of gravity with the axle, I 
think it would be far better to use a small counterpoise in the first instance; or 
even to consider the moment of the needle itself, (or its weight multiplied by the 
distance of its centre of gravity from the axle,) as a counterpoise acting with or 
against the magnetic moment. The ratio of this moment to that of the other coun- 
terpoise should, of course, be determined by‘ the indirect method which has been 
already explained.t 
We may now form an estimate of the accuracy of this method, as compared with 
the usual one, in the determination of the magnetic intensity. In the received me- 
thod, it is well known, the horizontal component of the force is determined by ob- 
serving the time of vibration of the needle suspended horizontally. Now let us sup- 
pose this portion of the force to be completely determined, and inquire how far the 
probable error in the dip will affect the total intensity, thence deduced. If h denote 
the horizontal component, we have 
h=9ocos 8; 
* We have here considered the effect of friction on the result, so far as it depends upon a single reading. 
The eight readings usually taken may undoubtedly diminish still further the resulting error of dip; but 
as these readings are taken in order to correct other errors, we have disregarded their effect here. If, how- 
ever, the needle be so perfect in its construction, that the errors arising from the non-coincidence of the 
centre of gravity with the axle, and the deviation of the magnetic axis from the axis of the needle, &c. 
are less than the error of friction, then the multiplication of readings will have the effect of reducing 
the latter in the ultimate result. 
f See page 6. 
