130 



PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE 



(3) and (7) First Tesseral Sine Harmonic and Sectorial Sine Harmonic. 

 These are defined thus : 



8 ' (/,) = cos e (/< 3 sin* e - 2 s ), 



= sin $ (tf - *' 3 cos 3 <), (s = l, 3), 



..... ( 54 ) 



where </ = i [2 + * 3 T (4 - xV 2 )*], with the upper sign for s = I, the lower for 

 s = 3, and g /3 = 1 </'. 



Writing 



i 3 = /r *, 



|| 3 ' ( M ) = /. K cos 6 (t* - /c 3 cos 3 6), 

 S' <, = . K- 1 sin < * 3 + *' 2 



(55) 



where /=, 9 = - 



Squaring ^ ;J J we find, 



^ (J = o, ,1, = t\ A,= - 2t\ A, = l. 

 On substitution in the formula for harmonics of the third order I find 



I 3 ' (sin) = 

 If we write 



f (K 3 - K' 2 ) KV 2 * 3 + | KV 4 ] ( - / 



* = * - ^' 3 ~ ' 6 (4 ~ *V 3 



^ 



* 



t 



- 8 - 34/c' 3 



3 i " (8 - 25/< /3 + 5 IK'* - 



On substitution 1 find, on putting f' z g~K~K rz = 1 , 

 r i / \ 47T& 3 cos 3 2 3 



+ ( K s _ K 't) ( 8 

 J 3 3 (sin) = the same with the sign of the square root reversed. 



. (56) 



