572 BEPOKT — 1891. 



makes the line a tangent to tbe conic 



Af + B + av''-2Fx-2Q.v!/ + 2ny = . . • . (2) 



If now a second line x — a = m'tj be obtained from the first by means of the 

 relation 



piii^ + 2f'mm' + qm''^ + 2q'm + 2p'm' + r = , , . (3) 



the envelope of this line will in general be of the fourth class. If, however, the 

 minors of the discriminants of (1) and (3) be connected by the relations 



ACQ ,.. 



P = E = Q' (^> 



then the envelope degenerates into two of the second class. 



The conditions (4j were obtained thus: — Considering a,m' ,ni as the coordinates 

 of a point referred to three orthogonal planes, (1) and (3) represent Uvo ct/lhiders, 

 and (4) gives the conditions that tliese should have two common plane sections. 

 The conditions may be geometrically interpreted thus ; if two conic cylinders lie 

 between tiuo parallel planes (i.e. each cylinder touch both planes) their complete 

 intersection corisists of tivo conies. 



Laguerre's transformation, analytically considered, gives the relation 



m- + 2knim' + m'~ = A:- — 1 . , . . (3') 



here 



P = 1-/^SR = 1-A^Q' = 0. 



Then by (4), A = C and G = 0, and the transformation is seen to be of simple use 

 only for the case of circles. M. Laguerre's results, however, of which an account 

 is given in his ' Geometric de Direction,' are of exceptional elegance. 



11. Note on the Normal to a Conic. By R. H. Pinkeeton. 



12. On the Importance of the Conception of Direction in Natural Philo- 

 sophy. By E. T. Dixon. 



This importance has already been recognised in the higher branches of science 

 in the guise of Vector theories, and the chief reason it has not been made use of in 

 elementary geometry is the want of a proper definition. Such proper definition 

 might be deduced from the conception of direction as a relation between two 

 positions which is independent of the distance between them and of the absolute 

 position of either of them in space. The concept thus defined is independent of 

 the conception of a straight line, and so may be used to define it, and is therefore 

 distinct from tbe concept defined, by saying that two straight lines which have a 

 common point have the same or different directions according as they coincide or 

 not. That some notion of direction is necessary to elementary geometry is shown 

 by the fact that without it right- and left-handed figures which are equal in every 

 respect cannot be distinguished; and that the concept as defined is commonly 

 entertained, is proved by the fact that it follows from Newton's Laws of Motion 

 that absolutely fixed directions may be conceived in space, although absolutely 

 fixed positions cannot.' 



' Vide The Foundations of Geometry (Deighton, Bell & Co.) 



