110 The Astronomer Royal on the Progress of Accuracy, 

 equation, 



and therefore (j/) s =Q J '. Similarly (^) z =q'. 



The expressions for p and q may be more conveniently put in 



/l 



{c-Jsa-7 — * ' — I . Si- a - - 



«^7 





P"=, T 



::sa — — —v — l.sina — - 



and 



W*M-i)H 



::5 3»+3*r— ^-l.ffli&e+Si 



{cos [ — 5 3a — oV) 



wMtJH 



-v -l.sin n-^.3a-f37r)} 



•:•:> - : :.-.-:- 



- v / _l. s iii(n-f/3".3a4-37r)}. 



In algebraic generality there is nothing to prevent E from 

 consisting of two terms, one being imaginary. But such an 

 expression could be put under the form of a cosine and a sine 

 with imaginary factor, and its effect would be simply to add a 

 constant to the constant $. and nothing would really be gained 

 in generality. And, upon attempting to solve the equations for 

 A„ and B„, it would be found immediately that the condition of 

 real values for A, and B„ requires that E' and E" be equal, and 

 thai 3' and ^ be equal. The equations are therefore to be used 

 in this form : — 



A.+y.B,=E.(?'/- 



A,+y.B,=E.( 5 y + <'. 



First, taking their difference, we find 



B =E : " £ - - ~ L 



" F-P" 



= -E. - hi h ---'- " • -■:: -Sa + frr) 

 2 ■ \ — 1 . SLn(flt-f ^") 



Second, multiplying the first equation by ]f=(q , ) t and the 



