294 OR- 8. CHAPMAN ON THE DIUKNAL VARIATIONS OF THE 



the equations, and some formulae derived from them, will be denoted by the same 

 Roman letters which lie uses.* 



i <r ......... (A) 



= (n+ < r)(+a-l)Q._ 1 *- 1 -(n-r+2){--H)Qt +1 (r -i (C) 



1 ' +1 ) ...... (D) 



81110 



1 '- 1 , . . . . . (E) 

 . . (F) 



sin 



+ (w+ 1) (n + cr) (w + o 1) Q,,,/- 1 }, 



Making use of these equations, we obtain the following expressions : 



In II.' ( p), the second term remains the same, while the expression in square 

 brackets in the first term becomes 



These expressions for R B * ( p) are of the type 



Now by equations (B) and (C), it is evident that Q/ + V~ l , Q/'V' 1 , and Q/y' can 

 be expressed as the sum of a number of tesseral harmonics all of type <r+p or all 

 of type <r-p (at will), and of degrees ranging, by steps of 2, from v(p-l), 

 v(p-l) and fp respectively. Further multiplication by y* can be so arranged as 



* Ibid., pp. 187-189. 





