On the Magnetic Effect of the Sun or Moon on Instruments. 295 

 in inverse powers of D, 



Hence, expanding, rejecting terms in which D -4 occurs, and 

 those into which coordinates of both dm and dm' do not enter 

 (since they would disappear in integrating, from the fundamental 

 property of magnetism that t \dm = G) } 



dY= + ~ry 3 — • (y z '—zy') + small terms, 

 dQ= H =p — . (zw 1 + 2xz f ) + rejected terms, 



.dU— j^ — . {2xiJ + ycd) + rejected terms. 



Let M and M' be the magnetic moments, and afiy, afft'y', 

 the directions of the magnetic axes, so that 



M = {§zdm) 2 + (§ydm)* + {§zdm)% 

 M' 2 = (pJni*)*+ ( ydm'Y+ i&irriY ; 



{xdm r. \y dm \zdm 



C ° Sa= M ' C ° S ^ = M ' C0Sry= -W~> 



. ^x'dm' nl {if dm! , ^z'dm' 



co * a= w' cos/3 m^ ^y-^-w-' 



Then integrating, the components of the moment turning the 

 needle round 0' will be 



P= + (cos /3 cos 7' — cos 7 cos /3') + small terms, 



MM' 

 Q= + ~tP~ ( cos *¥ cos a ' + ^ cos a cos y') + sma H terms, 



MM' 



R= ^p- (2 cos a cos /3' + cos ft cos a') + small terms. 



Hence the resultant moment tending to turn the needle round 0' 



MM' 



= -pv 3 \/ (cos /3 cos 7'— cos7Cos/S') 2 4-(cos7Cos«'4-2cos«cos7 / ) 2 



+ (2 cos « cos /3' + cos (S cos «') 2 + small terms. 

 The maximum value of this (neglecting the small terms) is 



