504 Archdeacon Pratt's Geometrical Proof of the 



obtain a definite picture, probably because a less energetic cor- 

 roding substance than water is necessary. 



Leydolt's microscopic observations of corroded surfaces have 

 indeed shown that crystals consist of molecules whose forms 

 belong to the crystalline series of the regularly built aggregate; 

 and the investigations of Volger and Scharff have proved that 

 the structure is manifold and complex : Brewster's luminous 

 figures show this in a still higher degree. How must a position 

 of the molecules and a difference in their particles be constituted, 

 which, as in calcite for example, produces a different result when 

 corroded with nitric acid to what it does when corroded with 

 hydrochloric acid ? and if it is not to be doubted that all the lines 

 of these figures indicate striations in directions at right angles 

 to them, what structure can produce the changes which vary 

 with each trace of a further corrosion, and the manifold curves 

 and spots which we perceive on the rhombohedral surfaces of 

 calcite and many other minerals corroded by nitric acid ? 



Theoretical crystallogenesis stands here, so to speak, before a 

 mirror which shows all the difficulties and puzzles which it must 

 overcome and solve ; and at present we cannot expect that it will 

 attain such a solution. Brewster said on this point, "In what- 

 ever way crystallographers shall succeed in accounting for the 

 various secondary forms of crystals, they are then only on the 

 threshold of their subject. The real constitution of crystals 

 would be still unknown ; and though the examination of these 

 bodies has been pretty diligently pursued, we can at this moment 

 form no adequate idea of the complex and beautiful organization 

 of these apparently simple structures." 



LXVII. Geometrical Proof of the fundamental principle of 

 Laplace's Functions. By Archdeacon J. H. Pratt*. 



1. 13 Y the fundamental principle of Laplace's Functions, I 

 J-J mean the property expressed by the formula 



) — iJo 



:47rF(/*,o>), 



{l + c*-2cpf 



where c=l, and where p, and co, and p! and at 1 are the polar 

 coordinates (measured as described below) to a fixed point Q and 

 a variable point P, situated on a sphere of radius = 1 about the 

 origin of coordinates, and p = the cosine of the angle subtended 

 by Q P at the origin. Let the diagram be so drawn that the 

 plane of the paper passes through the origin of coordinates and 

 the points Q and P. Let A be the pole on the sphere from 



* Communicated bv the Author. 



