August 30, 1900] 



NATURE 



429 



(2) Since the angle Y is equal to the supplement of the 

 angles at the base of the triangle pdd', p being the apex, the 

 orientation of the reflected pole (that is, the direction of the 

 north point of the field) is given by the equation 



ton i V cosi(p + 8) 



The law of rotation readily follows. The interval from the 

 passage of the star over the hour circle pd' being expressed by 

 /, with a day as the unit of time, -4 - « = znt, and the equation 

 becomes 



tan i Y = K tan i iitt. 



where K = ^"^i ('' + ') 



(6) When the reflected ray is in a horizontal and southerly 

 direction, as is usually the case, (i» = o, and p = ir- L, so that the 

 formula for orientation becomes 



cos \ (p - «) 



Hence :— (a) The rotation of the field has the same period as 

 the diurnal motion. 



{b) The motion is continuous and in the same direction, 

 direct or inverse according to the sign of K. 



(<:) The plane of reference is a plane of symmetry, since the 

 angle y has equal values of contrary sign at equidistant intervals 

 of time from passage across the reference plane. 



Prof. Cornu illustrates the rotation by a diagram similar to 

 those in Fig. 2. 



(3) The angular velocity of rotation at the epoch/ is given by 

 ds: _ K 



dt "" ^^ cos-ir/ + k*sin"^ir/ * 



where 



tan^ Y = Ktan J/4 



^_sini(L-8) 

 sini(L + 5)' 



It readily follows that there is no rotation of the field in this 

 case when the polar distance of the star observed is equal to the 

 latitude of the place of observation ; the rotation is clockwise if 

 the polar distance be less than the latitude, and contrary if 

 greater. Fig. ? illustrates the varying conditions of rotation in 

 the latitude of London (a) for the position of the sun at the 

 winter solstice, {b) for the position of the sun at the summer 

 solstice, and {c) for a star which passes through the zenith. In 

 each case the numbers are placed to represent the position 

 angles of the north point of the field at corresponding hour 

 angles. 



In the case of the heliostat, where the rays are reflected in a 

 northerly direction, a similar method of computation is adopted 

 by Prof. Cornu ; but as the instrument is so little used in work 

 of precision, it is unnecessary to give the details. The important 

 result is that the field of view under ordinary conditions has an 

 angular velocity of rotation greater than that of the diurnal 

 motion. 



Sun at winter solstice, London. 



Sun at summer solstice, London. 

 -Illustrating rotation of field of siderostat. 



Star through zenith, London. 



The denominator is always positive, so that the velocity has 

 always the same sign as K ; its value varies from 2irK (when 



2ir 

 / = o) to j^ (when / — \), and is equal to the diurnal motion 



when the conditions make the denominator equal to K. The 

 velocity varies so slowly for small values of t, that it may be 

 sufficient to regard it as constant and equal to 27rK. Since the 

 northern meridian passage cannot be observed with the side- 

 rostat, the value ^ is not observable. 



(4) The apparent motion of the field, as seen in the mirror, 

 will evidently be in the same direction as the apparent motion of 

 D V seen from outside the sphere, and it will not be reversed 

 by an astronomical telescope. When the polar distance of the 

 star observed is less than the supplement of the polar distance 

 of the reflected ray, the apparent direction of rotation of the 

 field of a siderostat is clockwise ; it is in the contrary direction 

 if the polar distance of the star is greater than this supplement. 



51 When cosi(p + 5) = o, we have K=o, and V = o for all 

 values of t. Hence there is no rotation of the field when the 

 polar distance of the star observed is equal to the supplement of 

 the polar distance of the direction of the reflected ray. 



In this case, p + S=i8o°, and pm = 9o°, so that the mirror is 

 parallel to the earth's axis, and the instrument thus behaves 

 like a coelostat. 



. J This gives the angle reckoned from the direction d'p. To obtain the 

 inchnation to a vertical line passing through the mirror, it would be neces- 

 sary to calculate the angle pd'z. 



NO. 1609, vol. 62] 



A knowledge of the orientation of the field as reflected by a 

 mirror is so frequently required that it may be useful to refer 

 briefly to other ways of treating the problem. 



Orbinsky proceeds much in the same manner as Prof. Cornu, 

 but considers the more general case in which the reflected rays 

 are neither in the meridian nor horizontal. The position of the 

 normal is midway between the direction of the star and that of 

 the reflected ray, on a great circle, so that the direction of the 

 reflected ray from any other point of the celestial sphere can at 

 once be determined. 



In this way the position of the zenith point of the field 

 (vertex) is derived with respect to the vertical circle in the 

 plane of the reflected ray. A calculation of the angle between 

 the vertex and the north point is then all that is required to 

 give the direction of the north point of the field with respect 

 to a vertical line through it. 



Another method of representing the orientation was adopted 

 by Mr. Shackleton in connection with the eclipse of 1896. 

 This can be applied to a reflection in any direction, but it will 

 suffice to indicate its application to a siderostat with the reflected 

 ray in the meridian. Using Prof. Cornu's notation so far as 

 possible, in Fig. 3 nesw is the horizftn, nps the meridian, p 

 the pole, D the star, M the mirror, MS the direction of the re- 

 flected ray from D, and sdn the trace of the plane of reflection. 

 Representing the direct field by anb, n is the north point. The 

 field of the mirror appears behind the mirror as a'n'b', a'b' re- 

 maining in the plane of reflection, and (5'n«' being equal to 

 bun. Since fiv is a vertical line through the field of the mirror, 

 and z'Na' = dsp, it is evident that van' = i8o° - (psd -f pds). 



