112 REPORT— 1898. 



in the value of Y, 



-JL(1 -/i^)-* m H,7 (g"^ sin m\-A;;' cos m\) 



and ?-"-Xl - /Lt2)-J m H^» (g-™,, sin m\ - A™„ cos to\) ; 

 in the value of Z, 



''^^^ H^ (5-^ cos m\+/C sin mX) and - wr"-' H;;' (g'"„ cos m,\ + /i™„ sin to\ 

 It is also proved tliat 



and {l-rY ""-^ = I (n-m) H--i-| (n+m) H» ^ ; 



and these relations are often useful in expressing the terms in the value 

 of X. 



It is found convenient to employ the notation with n and — n subscript 

 more generally to refer to internal and external forces respectively, and in 

 this sense the following notation is employed. 

 Let 



V™ = ~ H;r and V^-„ = r" H™, 



an 



diet 



be the coefficient of (^™ cos m\ + Jt^ sin mX) in the expression for X, the 

 force towards the north, and let X"„ be the corresponding coefficient of 

 (g"^„ cos m\ + /t!l„ sin mX) in the expression for X arising from forces outside 

 the Earth's surface. 



Then X!!!,, = r"-i [i (n-m) Hlf^i-^ (u + m) H,rl 



Using the notation Y"' and Y'H:^, and also Z,^ and Z!!.'„ in the same way 

 for the forces Y and Z, we have the potential 



V=2[V;r« cos m\ + /i™ sin to\)] + 2[V!?„(^::^„ cos mX+h"!,, sin mX)], 

 X=2[X'„"(j7^ cos mX+h^ sin mX)] + 2[X!!„(5?™„ cos toX+ A™„ sin ?nX)], 

 Y= 2[Y;:"(^;r sin mX - h^ cos mX)] + 2[ Y::?„(^!:?„ sin mX - A!:'„ cos mX)], 

 Z = 2[Z;r(5r:« COS m\ + /C sin m\)] + 2[Z!^„(^!!„ cos m\ + /i':'„ sin mX)]. 



Collecting coefficients of cos mX and sin mX in the values of V, X, Y and 

 Z respectively : 



The coefficient of cos to,\ in V is ^CV'^gn +^-n9-n), 



X „ 2(X-sr™ + X-,.^™„), 

 Y „ 2(Y-7C+Y!!„;i-„), 

 Z ^Z:'g^ + Zl„g1„). 





