AND THE MEAN SPECIFIC GRAVITY OF THE EARTH. 
597 
the south, but whether these may be sufficient to produce the effect observed muU be 
considered hereafter. 
Deflection caused hy an Attracting Mass. 
Let it be required to find the attraction exercised by a given mass placed on the 
surface of the earth upon a given point on the surface, the distance being supposed 
so small that the sphericity of the earth need not be considered. Let the position of 
any point of the attracting mass be determined by the coordinates r, d, z-, r and 6 
originating in the attracted point and being measured in the horizontal plane passing 
through that point, jz being measured perpendicular to this plane. Let also the value of 
^=0 correspond to the meridian line, then the volume of an indefinitely small element 
of the attracting mass being rdd.dr.dz, if ^ be its density, its attraction will be 
'.qdr .d^ .dz 
and therefore its attraction in the direction of the meridian is equal to this quantity 
multiplied by r.(r^+ 2 ^)**^-cos ^ ; so that the attraction of the whole mass is equal to 
cos Qdd 
A dr . dz 
{r'^ + zfl 
In order to perform, the integrations here indicated, the equation of the surface of the 
attracting mass is required to determine the limits ; this cannot be expressed, nor 
can which is also a function of rOz. But it is easy to find the attraction of a mass 
of uniform density included within the following surfaces : — The horizontal planes 
2 = 0, z — h, the two cylindrical surfaces defined by the equations r=ri, r=r 2 , 
being constants, and two vertical planes determined by the equations 0 — 6^, 6=0^, 0A 
being constants ; § being supposed also constant. Integrating between these limits, 
the attraction of the mass under consideration is found to be 
^9 + xX + 
A=oh(sm 0,-sm 0,) 
which being expanded is equal to (putting ri-{-r 2 — 2 r) 
A=Kr,-r,)(sin 0,-sin 
Hence, by taking sufficiently small. 
A=^(r 2 — ri)(sin 0 ^ — sin 
h 
or if £ be the angle of elevation of the centre point of the upper horizontal surface of 
the mass in question, at the attracted point 
A=^(r 2 — ri)(sin ^ 2 — sin ^ 1 ) sin £. 
If h be small, so that its square may be neglected. 
A 
=g'(sin ^2 — sin 0^ 
