with respect to the SEther. 419 



a r , y\ and W would be a very small fraction of 22 kilom. per 

 ■second, and accordingly equations (4) may be .simplified by 

 the omission of x\ y\ z' . 



6. These simplified equations can be treated by the method 

 of least squares. In this preliminary discussion, all the obser- 

 vations are taken to be of equal weight, so that we have 

 simply to make the sum of the squares of the left-hand 

 members of these equations a minimum with respect to a, b } 

 and c. Thus 



cil? +&Sfo + c2{J=V2f(VT-r)" 

 c&fri + bSflf +c?, v Z=VSv(VT-r) } 

 c&g+btvZ + ^? =V2f(W~ r) 



and from these equations values for a, l>, c can be at once 

 written down. 



7. Before discussing the choice of axes, it will be convenient 

 to obtain general expressions for the probable errors of a, b, 

 and c. The residuals Yr—r may be regarded as the sole 

 source of error, since all the other quantities involved in (5) 

 are known with ample accuracy for our purpose. In dealing 

 with the various errors which may affect the values adopted 

 for the quantities Yr—r, let the suffix distinguish true 

 values, while the suffix ., indicates those quantities which are 

 variable from one Yr—r to another. Thus let 



V=V +i>, r=r + kr ; .... (6) 



so that y is the error in the accepted value A" for the velocity 

 of light, and k is the proportional error in the value assigned 

 to the solar parallax. (The angular magnitudes involved in 

 our estimate of r can be so accurately ascertained that no 

 appreciable error can arise from them.) 



8. Again, let (t ) s denote the time actually taken by light 

 to travel from the satellite at mid-eclipse to the observer. 

 Then, (t ) a . being the difference between the apparent and the 

 actual time of mid-eclipse, the value r s which is available to 

 us will be liable to differ from (r ) s owing to three sources of 

 error : — 



(i.) The irregularity of Jupiter's surface causes the actual 

 eclipses to succeed one another at intervals of time which are 

 sensibly discrepant from those calculated theoretically. Thus 

 t, is affected by an error of prediction which is periodic or 

 quasi-periodic* in character, and which we denote by % s , 



* Though Prof. Sampson tells me that uo period has been recognized 

 in the residual errors. 



2 E 2 



