18 Prof. H. J. S. Smith on the Conditions of 



inquiries as I have liere made. I therefore submit tliem 

 frankly, though not di fid enthj, to the candid judgment of those 

 who are far better qualified than I pretend to be to pronounce 

 upon them. Perhaps the most startling thing I have advanced 

 is that heat is gained and lost in retarded expansions and con- 

 densations. But I would ask any thermodynamist how he 

 would explain, without this theory, the gain and loss of vis 

 viva in the rise of a barometrical column from one point of 

 rest to another, that vis viva being, of coarse, the measure of 

 the loss and gain of heat in the air that generates it. Be- 

 tween two points of rest the motion of the barometer 7mist be 

 at first accelerated, and it must be at last retarded ; and 

 between these two there must be some time when it rose 

 uniformly and the forces were in equilibrium. No vis viva 

 has been gained or lost on the whole ; but the air has been 

 all the while condensing. What, then, has become of the heat 

 generated? 



III. On the Conditions of Perjjeyidicularity in a ParaUele- 

 pipedal System. By H. J. S. Smith, FM.S., Savilian 

 Professor of Geometry in the University of Oxford^ . 



1. rriHE conception of a parallelepipedal system {i. e, of a 

 -«- space divided by three systems of equidistant parallel 

 planes into similar and equal parallelepipeds) may be regarded 

 as forming the basis of the usually received theory of crystal- 

 lography. It is the object of the present note to state some 

 of the conditions for the perpendicularity of lines and planes 

 in such a system. The results of this inquiry (which has been 

 undertaken at the request of Professor N. S. Maskelyne, and 

 owes much to his suggestions) are submitted to the Crystallo- 

 logical Society with great diffidence, because they do not 

 seem likely to admit of any direct application to the practical 

 work of the crystallographer. Such interest as they possess 

 belongs to a domain which borders on the one hand on pure 

 arithmetic, and on the other hand on pure geometry. 



2. It is perhaps hardly necessary to explain that by a ^^line 

 of the system " we understand a line joining any two points of 

 the given parallelepipedal system, by ''' a plane of the system " 

 a plane containing three points of the system, the points of 

 the system being the points of intersection of the three sets of 

 equidistant parallel planes by which the system is defined. 

 It will be sufficient to consider oriyin-MnQs and planes^ ^. e. 



* Communicated by the Author, having been read at the Meeting of 

 the Crystallological Society, June 14, 1876. 



