402 SEE— THE NEW THEORY OF EARTHQUAKES [November .s, 



and the resulting force is given by the diagonal of the parallelo- 

 pipedon 



/= l//f +r;^+-r. (3) 



If the direction and intensity of the force is found by observation 

 all the desired elements may be calculated. 



Passing to polar coordinates Gauss denotes by r the distance of 

 the point from the center of the earth, and by u the angle which r 

 makes with the north polar axis, and by A the angle with the meridian 

 made by a plane passing through the point and this polar axis. 

 Symbols with subscripts zero refer to points in the earth. 



Hence the expression for the distnce p from any point in space is 



p^ = r^ — 2rr^ [cos u cos u^ + sin u sin ti^ cos (A, — X^,)] + r^. (4) 

 Accordingly the expression for the potential becomes 



V f ^^ (5) 



J Vr^ — 2rr^ [cos u cos u^ + sin u sin u^ cos (\ — \)] + ^ 



Considering the earth to be a sphere of radius a Gauss puts 



-F=iV + jV + jV" + ^>"'+--S (6) 



the coefficients, P^ P' , P" , etc., are functions of A and u only. 

 If we develop the denominator under the integral sign in (5) 

 in a series, the first two members of the series become 



- + -| [cos u cos u^ + sin ii sin u^ cos (A — AJ] 4- 4 + • • •, (7) 



whence 



^2po ^ __ J*^^^ ^zpt ^ _ {a cos ?^ + yS cos A + 7 sin « sin A} , 



where 



a =fr^dfJL cos z/^, jS =^r^dii sin u^ cos A^^, 7 =Jr^^dfj> sin ?^ sin A. (8) 



As Gauss took the northern and southern magnetism to be equal 

 and of opposite sign, it follows that 



p/x = 0, 



