GRAVITATIONAL STABILITY OF THE EARTH. 233 



57. The surface harmonics of the first degree expressed in ordinary spherical polar 



co-ordinates 6, d> are . /, . /, , 



sin cos <p, sm sin <f>, cos ; 



and any spherical surface harmonic of the first degree can be expressed in the form 



(p cos< + </ sin <f>) sin 0+r cos 0, (115) 



where p, q, r are numbers. The spherical surface harmonics of the second degree 



' I 1't ' * 



sin 20 cos <f>, sin 20 sin <f>, sin*0cos2<, sin* sin 2<, 3 cos 20+1; 

 and any spherical surface harmonic of the second degree can be expressed in the form 

 (acos< + y8sin<)sin 20+ (y cos 2^ + Ssin 2<) sin* 0+e(3 cos 20+ 1). . (116) 



The spherical surface harmonics of the third degree are 

 (i.) The zonal harmonic g ^ _^ ^ Q . 



(ii.) The tesseral harmonics of the first rank 



(5 cos 2 6 1 ) sin cos <, (5 cos 3 l)sin 0sin <f>; 

 (iii.) The tesseral harmonics of the second rank 



sin" 6 cos 6 cos 2$, sin* cos sin 2< ; 

 (iv.) The sectorial harmonics 



sin" cos 3<f>, sin 3 sin 3<f>. 

 Since 



5 cos s 0-3 cos = | (cos 30+ cos 0), 

 (5 cos 3 0-1) sin = \ (sin 0+ 5 sin 30), 

 sin* cos = \ (cos 0- cos 30), 



sin 8 = \ (3 sin 0- sin 30), 

 any spherical surface harmonic of the third degree can be expressed in the form 



aW + X(hcoa<f> + c8in<f>)+Y(dcos2<t> + esin2<f>)+Z(fcos3<l> + (j8m3<l>), . (117) 

 where n, b, c, d, e, f, g are numbers, and 



W = cos 30+ (0-6) cos 0, I 

 X = sin 0+5 sin 30, 



Y = cos 0- cos 30, [ (118) 



Z = 3 sin 0- sin 30. 



1 The form 3 cos 20 + 1 for the zonal harmonic is 4 (| cos* - J), and is token aa teing more convenient 

 for calculation. 



VOL. CCVII. A. 2 H 



