The Rev. Dr. Robinson on the Longitude of the Armagh Observatory. 1 13 



the two comparisons, and a its rate, (+ when losing, because it increases the 

 positive correction,) we obviously have 



L = E— w + eXi, 



and accenting the letters for the return, 



l = e' — w' — r'Xi'. 



If we suppose k = r', that is, either the rate unchanged on the road, or similarly 

 disturbed in the two journeys, then we have 



^ _ (e'-w')-(e-w) ^j^ 



i + i' 



which may be called the travelling rate, and is given by subtracting from the 

 watches' change between the two eastern comparisons the change between the 

 two western, and dividing by the difference of the intervals; and this obviously is 

 the rate which should be used. 

 We have also 



2l = e'— w' + E — w-|-rX(i — (2) 



from which it is obvious, that if the times employed in going and returning are 

 equal, or nearly equal, the effect of an error in the assumed rate is insensible in 

 the mean of the two. 



As the expression of r assumes that the longitudes obtained going and 

 returning are equal, it is obvious that when the travelling rate is applied, it is 

 useless to compute them separately. 



If we suppose that e — w requires a correction e, whether caused by errors in 

 the comparisons, or by accidental disturbance on the journey, then we obtain a 

 value of R by eq. (1), which requires the correction 



'' ~ I + 1' 

 and the correction of the mean longitude given by eq. (2) 



I + I' 



which in general will differ but little from that which occurs if we use stationary 

 rates, 



VOL. XIX. Q 



