108 The Astronomer Royal on the Progress of Accuracy, 



I shall not delay longer on this first operation, except to make 

 the following remark. At the end of this first operation, the 

 new values of A and C are equal but with opposite signs ; and 

 it is evident that the same remark will apply to the result of 

 every succeeding operation. But it did not apply to the values 

 A, C with which we started. We must therefore consider these 

 values obtained after the first operation as the beginning of the 

 symbolical series; and we may call them A , B , — A . 



Suppose now that we have gone through n operations, forming 

 the values A n) B n , —A n ; and that we examine the effect of tbe 

 ra + lUh operation. We have 



Before rubbing . A, 



After first rub . + \A n 

 After second rub. + ^A n 

 After third rub, 

 completing the I _j_.3A, 

 n+ l)th ope- 

 ration . . 



Which are the" 

 same as 



We have therefore the equations 





8 D « 



B w 



1\ j. IB 



+ iA n +|B„ 

 + iA„ + |B n 



. -A. 



-K 



-iA.-iB. 

 -&A.+4B, 



— A n 



+ 1 



A n +i — + 

 B w+ i= + 



A - X B 



n "c ±J nt 



-A w +--B n . 



Multiply the second by an indeterminate constant^, and add it 

 to the first ; then 



(A n+1+P . B„ +1 ) = (|+ Ij,)A. + (-|+ -1 -p)B„ 



Determine p so that — - — ~- =p ; and put q for the correspond- 



3 1 



ing value of -g- + -±p. Then the equation is 



(A n+1 +p . B n+1 ) = g . (A n +p . B n ), 



of which the solution is 



A„+/>.B„=E.^; 



where E is a constant, to be determined so as to satisfy initial 

 circumstances, and where j3 may be fractional. 



As the equation for p will be a quadratic, there will be two 



