AND THE MEAN SPECIFIC GRAVITY OF THE EARTH. 
603 
The probable error of x depends on the probable errors of a' — b', b' — d, and c'—a', 
that is supposing the geodetic amplitudes to be free of error, on the probable errors 
of (Pi and p 2 . The part of x involving p, and Pg is 97!^ X (13’037p,+ 1 TO 7 OP 2 ): con- 
sequently the probable error oi x is equal to the probable error of •1343(Pi-)-0’85p2), 
which, by means of the expression given for the probable error of Pi+i ^?’25 becomes 
(making jW»=0'85) 
probable error of j;= + 0'0053. 
Mean Density of the Earth. 
Having now ascertained the ratio of the mean density of Arthur’s Seat to the 
mean density of the earth, the knowledge of the latter results immediately from the 
knowledge of the former. Assuming as the result of observation 2’75 for the mean 
densitv of Arthur’s Seat, it follows that 
2'75 
Mean density of earth =.-^y^=:5‘316. 
The probable error of the divisor '5173 being '0053, the probable error of the resulting 
mean density is +'054, so that, considering no other cause of error than those of the 
zenith sector observations, we have 
Mean density of earth =5'316+'054. 
The General Defection. 
We proceed now to examine into the question of the sufficiency of the cause before 
mentioned, namely, the defect of matter to the north of Edinburgh and the accumu- 
lation of matter to the south, to produce the general deflection that is observed to 
the amount of 5", or rather more. In the first place, let it be required 
to find the attraction of a rectangular film ABCD, whose thickness is 
h and density g>, upon a point P in the production of one of its sides, 
AD. Measure x along PA and y perpendicular to it in the plane of the 
rectangle, then the mass of a small element is ^hdxdy, and therefore its 
attraction in the direction AP is 
qhxdxdy 
the integral with respect to x between the limits aa! is 
dy dy 
.{(d+y^Y {a'^+y' 
which being integrated from y=-0 to y=.h, is 
a! 
