xiv PREFACE TO PART II. 



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

 (g n cos m\ + h n sin m\) in the expression for X arising from forces outside 

 the Earth's surface. 



Then X~ - r" - ' [ (n - m) H +1 -%( n 



Using the notation Y and Y'" n , and also Z and Z, in the same 

 way for the forces Y and Z, we have the potential 



V= 2 [ VI (g: cos roX + /C sin mX)] + 2 [ F? B (#!' cos m\ + h m n sin m\)] 

 Z = S [X;r (#; cos HiX + /C sin wiX)] + 2 f JTr, (#! cos mX + /C, sin wX)] 

 r = 2 [ r;; (^ sin mX - /* cos mX)] + 2 [ 3T. (fl', sin mX - h? H cos wiX)] 

 Z= 2 [ Z;; 1 ((/;;' cos n\ + A sin mX)] + 2 [ Z!' B (^ cos mX + h\ sin mX)]. 



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

 respectively : 



The coefficient of cosmX in V is 2(F"V + F" 1 o* ) 



\ n & n n tj n I ' 



V "*t / \ r 'f, m I V'n ,.m \ 



A " ^( A ^9n +^-u9-n)> 



rv / y""7,7n i Y m L m \ 

 *\.-* B " + * -'*_), 



7 V i r / m r, m I V ' ^, \ 



" ^ * (^ M <7 n +4^g_ n ). 



The coefficient of sin ?>iX in F is 2 (F;"/i,+ F B A" n ), 



I 7 " V I V' 7, ' i V lm \ 

 ' A " *(-^n ^ + -A --), 



T^ 7 " V / V 7 "'" y* m I X 7 ""* \ 



1 " ^ l^nfl'i. + -*-#-), 

 y^ 5 1 /7'" /]'" _L X" 1 7i M \ 



^ " ^W.. /l + Z - re ft- H ), 



in which n takes all integral values for a given value of m. 



In a portion of his work, in which he treats of the definite integral of 

 the product of two Legendre's coefficients, Professor Adams proves the well- 

 known formula? that when n and n, are different from one another 



-1 

 and that when n 1 = n, 



