292 DR- S. CHAPMAN ON THE DIURNAL VARIATIONS OF THE 



lini,' tin- i-xpressions for X, Y in terms of the velocity potential \Js, the left- 

 hand side of the last equation, after division by Ga sin 6, may be written* 





 27+l 



2(2m + l) 



+ {- (m-1) (ro+ 1) (ro + r) (ro + r-l) Q,,,-! 7 - 1 



The right-hand side of our equation for R, after division by Ga sin 0, becomes 

 equal to 



K ! d* a d a d 



^ ysl y 



We suppose R to be expressed as the sum of a number of tesseral harmonics 

 p H "ty n sin (o-V a'), where p n " is a numerical coefficient, X' has been written for X + , 

 and & i^anges (possibly) from oo to +00. The contribution of each such term to the 

 total value of the last expression is easily seen to be the product ofp n " into 



-n (n+ 1) Q/ sin (erX'-a') {/ + 22/ p cosj>X'}, 



where we have inserted the values of K and its differential coefficients, and have 

 transformed the first line by means of LAPLACE'S equation 



iSo far f f and f p have been defined only for positive and zero values of p ; we now 

 extend the definition by the equations 



JP J -pi J p = j - p - 

 * Ibid., pp. 188, 189. 



