with respect to the ^Ether. 42 & 



of the coefficient of c in each of the first two ol equations (5) ; 

 these two equations being then solved for a and b. 



16. In the simplified problem of § 12, it appears that, 

 without special weighting of the observations, the probable 

 error in the velocity-component b can be greatly reduced by 

 a slight change of axes. The axis of x remaining unchanged, 

 the axes of y and z are to be rotated in their own plane 

 through a small angle ; and if the velocity-components along 

 the new axes are called a, b u c lt it is found that the probable 

 error in b x is a minimum when the axis of y lies very nearly 

 in the plane of Jupiter's (modified) orbit. If we take that 

 plane as the plane of xy, the probable errors, corresponding 

 to our simplified problem, are : 



p.e. in a = 44'0 kilom. per second^ 

 „ „ &i = 45'5 „ „ V ; . . (12) 



„ „ c 1 = 10,000 „ „ J 



using 330 observations, each with a probable error of 4%> 

 seconds. 



17. In the absence of definite reasons to the contrary, it 

 would be natural to take the plane of the ecliptic as one of 

 the coordinate pianos, but it appears that, by taking the plane 

 of Jupiter s orbit instead, we can avoid increasing the pro- 

 bable error of one velocity-component (b or bi) by an unknown 

 amount, and this without adding seriously to the labour of 

 computation. In the actual problem, therefore, it will pro- 

 bably be advisable to determine the values of two velocity- 

 components in the plane of Jupiter's orbit, using for this 

 purpose equations corresponding to the first two of (5), and 

 causing the coefficients Xff, S^f to disappear as nearly as 

 may >eem necessary by a moderate special weighting of the 

 observations. We may then proceed more tentatively, on 

 the assumption that c is moderate, to identify the velocity- 

 component b Y so found with the corresponding component b 

 in the plane of the ecliptic. If, for example, on general 

 grounds, and in the absence of any positive information, we 

 assume <? = 0±500 kilometres per second, the probable error 

 in J> thus arising will be approximately 12 kilom. per second,, 

 and the total probable error in b will be about V (45"5 2 + 12 2 ) 

 = 17 kilom. per second. 



I hope very shortly to consider this problem more in 

 detail ; meanwhile I wish to thank Prof. Sampson for his 

 kind and most helpful advice. 



Boar's Hill, near Oxford, 

 9 October, 1909. 



