112 Mr, Herschel's account of a series of observations 



Consequently, still neglecting the same things we get 



C— Z"— y {P+Q+C— E} 



for the difference of the timekeepers C and Z reduced to the 



epoch, and putting 



R= mean of all the (C - Z") - y.mean of all the (P+Q+C— E) 



R will be their most probable difference reduced to the fixed 



epoch. 



P, Q, and R, being thus obtained, we must obviously have 

 for the correct difference of longitudes, 



A = P + Q+R. 

 Now, substituting for P, Q, R, their values, this gives 

 A=mean of (A — B) + mean of ( B' — C) + mean of(C' — Z") 

 -{-^.mean of {A — E) 

 + {y^P).mean of {P + B'— E) 

 — y.mean 0/ (P + Q + C" — E) 

 that is, reducing, 



A =zmean of (A— B) + mean o/(B'— C) + mean o/(C"— Z") 



+ (3. mean of A-\' {y — fi), mean ofB' — y, mean of C 



_P/3_Qy. 



This value of A is however susceptible of still further 



reduction by substituting for P and Q their values ; which if 



done, and the powers and products of (3 and y neglected, as 



has all along been done, we get 



A = mean of {A — B) + mean o/(B'— C) + mean of (C'^Z") 

 + ^.mean of A + (y — jS ) mean of B'— y. mean of C" 

 — /3 . mean of (A — B) — y. mean of (B' — C) 

 that is, finally (since the numbers of the observations of 

 A and of B are necessarily equal, and therefore the mean of 

 the values of A — B is equal to the mean of A — the mean 



