112 



PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE 



involved, but the notation used in the two later papers seems preferable where the 

 formulae are rigorous and symmetrical. 



In " Harmonics " the squares of the semi-axes of the ellipsoid were 



o 7 o i a 1 ~T~ /J \ 7 2 70, / ^ i\ 2 722 



a" = "( i/" ---- f>}> = * (" ~ 1)> c = k " 





The rectangular coordinates were connected with ellipsoidal coordinates v, p,, <f> by 



x~ 



2 



The three roots of the cubic 





=1 



were 



u 



/ -> 2 7,2 2 I." 1 ft COS 



- K V , Mo AT/A", M 3 ft" ~y_ o 



Lastly v ranges from co to 0, /n between i 1, from to ^TT. 

 In the two later papers 1 put 



, 

 1 + /8 



K sin y 



= sin 



and for convenience I introduced an auxiliary constant /3 (easily distinguishable from 

 the ft of the previous notation) defined by sin ft = K sin y. 

 The squares of the semi-axes of the ellipsoid were then 



. : _ k~ cos 3 y ,. 2 _ AT cos 2 ft o _ k' 

 sin 2 ft ~sm3"/8" ~ sin 2 /3 ' 



The rectangular coordinates became 



The roots of the cubic were 



7. '2 



This is the notation which will be used in the present paper. 



F 



K* 



cos 



