132 PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE 



In " Harmonics " I defined 



= 15 cos 0[- (1 - - 



In order that this may agree with our former definition we must take 

 /.= - 15(1 -- W + tf/3 2 ), /.</* = -- 15(1 - f/3 + 



Whence /* = -- 15 (1+ P - f^S 9 ). 

 It will be found that 



(60). 



so that / K . '/ 3 has the correct form. 



Again I defined 



(61). 



In order to make our former definition agree with this we must take 



!/ K ' \/ 1 . 2 3 (1 -|- .I,/? ssfiP)) f/ K '\/ 1 K/2 4. 



Therefore 0V - 1 . = - -*, . 



It will be found that t~ = %K'~ (I -f ^ a 2 -/8 2 ), so that ^r/c'v/ 1 . t* has the 

 correct form. 



Then 



345 



and 



Whence 



^ VV 3 = 33 - 42 -5 3 



K* 



(62), 



agreeing with the result on p. 548 of " Harmonics" with i = 3, s 3, type OOS. 



(5) The Second Tesseral Sine Harmonic. 

 This is defined thus : 

 P s s (/i) = sin 6 cos (1 -K 2 sin 2 0)* =/. K cos d (K"-K Z cos 2 6) 5 (/c' 2 +/c 2 cos 2 0)* 1 



*A*M*-**H r (63) 



where /= , ^ = 



