206 OBSERVATIONS OF THE TERRESTRIAL 



When ju - juo is so small as it is within the limits of the present 

 district of observation, we may take 



sin GU - /i ) = M - Mo> cos (ju - jtio) = 1, sin |3 = 0, 

 and the preceding equations become 



|3 = (ft - jUo) COS X. 



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 0, making the 

 angle u with the meridian ; then, in the same order of approxima- 

 tion, the perpendicular from the point M upon that line will be 



p = |3 cos u - a sin u ; 

 and, substituting for a and /3 their values just obtained, 



p = (n - no) cos X cos u - (A - Ao) 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 obser- 

 vation. The increase of the force, or of the dip, may (throughout 

 the limited area of the present district of observation) be assumed 

 to be proportional to the distance, measured in a direction perpen- 

 dicular to the line of equal force, or of equal dip. Accordingly, if 

 u be the angle which the line of equal horizontal intensity passing 

 through makes with the meridian of the place, the difference of 

 the intensities at the two stations will be proportional to p, or 



h - h<> = rp ; 



h and A being the horizontal intensities at the two stations, and r 

 a constant coefficient which determines the rate of increase. Sub- 

 stituting, then, for p its value (C), and making 



r cos u = x, r sin u = ?/, (D) 



we have 



(ju - ju ) cos \ x - (\ - Ao) y = h - h Q . (E) 



The equations of condition deduced from the observations of total 

 intensity, and of dip, will be of a similar form ; and the coefficients 

 of the unknown quantities, in the first member of the equations, 

 will be the same. 



