Feb. 28, 1889] 



NATURE 



423 



Investigation, I will proceed at once to the general method fo^ 

 otermining aberration, which M. Lcewy discusses after treating 

 ■me special cases. 

 The determination of aberration requires the measurement of 

 tlie distance between a pair of stars at successive epochs when 

 the effect of aberration on the angular distance is reversed. The 

 bservations are made when the two stars have the same altitude, 

 1 that the effect of refraction is a minimum, and the comparison 

 i the two measures gives a multiple of the constant of aberra- 

 tion, which is independent of all instrumental errors and also of 

 jirecession and nutation, as the distance between two stars is 

 unaffected by any movements of the earth's axi'^ or of the ecliptic. 

 There is the further advantage in the new method, that the effect 

 of aberration as measured is much greater than in the ordinary 

 methods of observation. 



But the result might be affected by change of refraction or by 

 alteration in the angles of the double mirror resulting from 

 thermal expansion between the two epochs of observation, and 

 ^[. Loewy has therefore imagined a general method of obser- 

 vation which eliminates any possible effects of the kind, as well 

 as methods applicable to special cases which determine any 

 changes due to refraction or expansion of the mirror. 



The essence of the general method is that two pairs of stars 

 are observed, the four stars being selected so that at the time of 

 observation they are all simultaneously at the same altitude, and 

 that the effects of aberration on the two arcs connecting the 

 stars of each pair are large and of opposite sign. Thus the 

 two arcs formed respectively by the two pairs of stars are 

 compared simultaneously both at the first and at the second 

 epochs. 



The first point for investigation is the effect of aberration on 

 the angular distance between a given pair of stars. From the 

 geometrical conditions, M. Lcewy arrives readily at the result 

 that the effect is proportional to the cosine of the angle be- 

 tween the median ^ of the arc and the direction of the earth's 

 motion. 



Calling A the angular distance between two stars, p the angle 

 lietween the median of the arc joining them and the direction of 

 the earth's motion, and k the coefficient of aberration,, the effect 

 of aberration is given by the formula — 



dh. 



■2k sin — cos/. 

 2 



It readily follows from this that the effect of aberration on 

 the difference of the two arcs connecting two pairs of stars 

 will be greatest when the two medians are on the same vertical 

 circle on opposite sides of the zenith. Under these circum- 

 stances, the effect of aberration on the difference of the two arcs 

 is equal to 



, . A . A' , 



4^ sm — sm — cos L, 



2 2 



A' being the angular distance between the two medians, and L 

 the angle between the direction of the earth's motion and the 

 line of intersection of the vertical plane through the medians 

 with the horizon. Thus the effect is proportional to the cosine 

 of this angle, and the greatest effect will be obtained when the 

 vertical plane of the medians, the ecliptic and the horizon inter- 

 sect in the same line, and the observations are made at the two 

 epochs six months apart when the direction of the earth's motion 

 coincides with this line, L having the values o^ and 180° at the 

 two epochs respectively. In that case the effect of aberration on 

 the difference of the two arcs has opposite signs at the two 

 epochs, and the comparison of the two sets of measures of the 

 two arcs gives 



where E is the difference of the two measures of difference of 

 arcs at the first and second epochs respectively. 



The next point for consideration is the choice of the angle for 

 the double mirror, their angular distance (A) between the two 

 stars in each pair being necessarily twice this angle. Obviously 

 the altitude at which the observation of the four stars is made 

 diminishes as A and A' increase, and M. Loewy shows that the 

 maximum effect at any given altitude is obtained by making 

 a'=: A, or the angular distance between the medians the same as 



' The median is the line bisecting the angle between the directions of the 

 two stars. 



that between the two stars in each pair. He then gives the 

 following table of the altitude h and of the effect of aberration 



— ' corresponding to the several values of the angle of the double 

 k 

 mirror a : — 



a 30" 35° 40° 45° 50° 55" 60° 



h 48" 35' 42' 9' 35° 58' 30' or 24° 24' 19" 12' 14 29 



f 20 26 3-3 40 4*7 5'4 6-0 



K 



M. Loewy concludes that the angle of the double mirror 

 should not exceed 50°, and he considers that, on the whole, it 

 would be well to make it 45°, so that the altitude of the stars 

 would be 30°, and the angular distance for each pair 90°. Under 

 these conditions, observations made at two epochs six months 

 apart would give as the quantity measured four times the con- 

 stant of aberration, while the ordinary methods of observation 

 only give at the maximum a measure of twice the constant. But, 

 in order to avoid daylight observations, M. Lcewy thinks it 

 would be advisable to be satisfied with a slightly smaller co- 

 efficient of k (the constant of aberration), say three instead of 

 four, which would reduce the interval between the two epochs 

 to about ninety-eight days ; and, by combining the observations 

 in the first five weeks with those in the last five, a series of 

 equations would be obtained, in which the coefficient of -^ would 

 vary from three to one, the mean value being about two. All 

 the observations could then be made in the night hours. 



Besides the general method of observation just described, M. 

 Lcewy has, as already mentioned, devised two methods applic- 

 able to special cases which are well suited to give independent 

 determinations of the constant of aberration. 



The first method consists in the observation of two pairs of 

 stars, of which one pair gives, at the end of two or three months, 

 the measure of twice the constant of aberration, and the other, 

 completely unaffected by aberration, exhibits the effect of tem- 

 perature on the double mirror. The first pair of stars should 

 be in the neighbourhood of the ecliptic ; the second pair is, as 

 will be seen from geometrical considerations, to be chosen so 

 that the latitudes of the two stars are the same, and that their 

 longitudes differ by 180°, in order that the arc joining thera may 

 be unaffected by aberration. 



This method is, however, not applicable at observatories 

 within 20° of the equator, and on this account, as well as to 

 give another independent determination of the constant of aber- 

 ration, M. Loewy proposes a second method, according to which 

 the angular distance of a single pair of stars near the ecliptic is 

 to be observed for a period of three months or longer, the mea- 

 sures in the first and last twenty-five days of the period being 

 used to determine the aberration, and those in the intermediate 

 forty days to deduce the effect of temperature on the double 

 mirror. 



The question of the adjustment of the double mirror remains 

 to be mentioned. This must be mounted so as to turn about the 

 optical axis, and this axis should coincide nearly with the axis 

 of figure. The effects of any movements of the double mirror 

 will then be as follow : — 



(i) In turning round the axis of figure the two images are dis- 

 placed in opposite directions, but perpendicularly to the trace 

 of the common plane of reflection. 



(2) In turning round an axis in this plane and perpendicular 

 to the axis of figure the two images move in the same direction 

 perpendicularly to the trace of the plane of reflection. 



(3) If the double mirror turns about an axis perpendicular to 

 the plane of reflection, the two images nK>ve along the trace 

 without changing their relative distance. 



Reference has already been made to the applicability of M. 

 Loewy's new method to the determination of refraction at various 

 altitudes. This was, in fact, the immediate object which 

 M. Loewy had in view when he devised the method, and his 

 investigation of the conditions of the problem was communicated 

 to the French Academic des Sciences early in 1886, the year 

 before he published his memoir on aberration. 



In his series of papers on the determination of refraction pub- 

 lished in the Comptes rendus, vol. cii., M. Lcewy first gives a 

 method for determining the constant of refraction, the law 

 according to which refraction varies with the altitude being 

 known. A pair of stars is observed when refraction has its 

 maximum effect on their angular distance, and again when the 

 effect of refraction is a minimum. For the maximum effect one 

 of the stars must be on the horizon, and the other in the same 



