Februaey 14, 1902.] 



SCIENCE. 



263 



c is the constant connecting density and 

 pressure in Laplace's celebrated hypothesis. 

 Assuming the density to vanish at a dis- 

 tance r from the center of such a body, it 

 turns out that V, p and p are given by 

 the following formulas, in which p^ and p 

 are the central density and pressure, re- 

 spectively, and M is the mass of the star: 



a =:7r/rj, g = sin ar/(ac), 



V=^(l- 



f = p<=q, p = 'PcT. 



The energy of condensation of such a 

 mass is found to be 



ri'k 



\Fr,, 



where P is the force of attraction between 

 M and an equal mass of infinitesimal vol- 

 ume situated at a distance r,, from the 

 center of M. It will be observed that the 

 results here given require no hypothesis as 

 to the temperature of such bodies. 



Optical Distortion of Photographic Tele- 

 scopes: Haeold Jacoby. 

 The observations discussed in the present 

 paper form part of a more extended series 

 undertaken in the year 1895, having for its 

 principal object a study of the optical dis- 

 tortion of astronomical photographic ob- 

 jectives. A question had been raised as to 

 the fidelity with which photographic tele- 

 scopes reproduce upon the negative exact 

 relative positions of the stars as they ap- 

 pear on the sky. This matter is funda- 

 mental to the art of astronomical photog- 

 raphy throughout the entire range of its 

 more important applications to stellar 

 parallax, interstellar motion within the close 

 clusters, and star charting in general; so 

 that the large amount of labor involved 

 even in its partial elucidation does not ap- 

 pear to be superfluous. Valuable coopera- 

 tion in the work has been granted with 

 ready kindness by several astronomers ; and 

 with their aid the special problem under 



investigation has been solved in a fairly 

 satisfactory manner. 



This special problem may be thus 

 stated: Is the scale- value of an astro- 

 nomical photograph absolutely independ- 

 ent of the direction of measurement 

 on the negative? In other words, if we 

 determine the coordinates of star-images 

 on the plate in millimeters with reference to 

 a pair of rectangular axes, the question is : 

 Will a distance of one millimeter measured 

 from the center of the plate along the X-axis 

 correspond to precisely the same number of 

 seconds of arc on the sky as a distance of 

 one millimeter measured from the same 

 center along the Y-axis? The matter may 

 be put in still another way. Suppose there 

 were upon the sky a number of stars so 

 situated as to form a small but perfectly 

 exact circle. Would a photograph show 

 these stars situated upon a similar exact 

 circle on the negative, or would defects of 

 the object glass distort their position into 

 an ellipse-like oval, after the manner of 

 atmospheric refraction? If this is the 

 case, equal diameters of the circle on the 

 sky will become unequal diameters of the 

 oval on the plate; and, in general, equal 

 distances upon the sl^ expressed in sec- 

 onds of arc will become unequal distances 

 upon the plate expressed in millimeters, 

 even after correction for all knoAvn sources 

 of difilerence, such as refraction, aberration, 

 etc. 



Various investigations of optical dis- 

 tortion have been published by Donner, 

 Turner and others ; but they were all made 

 by methods necessitating a knowledge of 

 relative star positions based on measures 

 other than photographic. To avoid this 

 inherent difficulty, the writer suggested in 

 1893 a process in which it is not essential 

 to have a precise previous knowledge of 

 relative star positions. It is thus rendered 

 entirely unnecessary either to make a labo- 

 rious and time-consiiming heliometer trian- 



