a theorem of Dirichlet 115 



It is evident that any rational point (*•, y, z) in space of three 

 dimensions can be regarded as being determined by its affix 

 V^x-]ry^ + z^'\ where ^ is the real root of the equation ^^ = n : 

 also the affix of any point determines a plane through that point 

 and parallel to the asymptotic plane of the surface whose equation 

 is i\=(jc? -\- ny"^ + n"z^ — Snxyz = 1 ; such a plane we shall call a 

 " r-plane ". 



We shall now prove the following proposition : 



The V-planes of any two consecutive integj^al points on the 

 surface A = 1, together with the surface itself, enclose a space of 

 constant volume. 



The equation A = 1 can be written in the form 



[x + y^ + 2^-| [{x - y'^y + (2/^ - z^^-'f + {z"^^ - xy^ = 2 ; 



so that the section by the F-plane of the point (^, rj, f) will be 

 given by the equations 



x" + 2/-^2 ^ ,^^^.^2 _ ,^^y^ _ ^2^^. _ ^^.y ^ ijY (i) 



and a; + 2/^ + z"^'- = T. 



Evidently the quadric (i) and the surftxce A = 1 are cut in a 

 common section by the F-plane of the point (^, 77, ^). It is this 

 quadric that we shall now examine. 



If by any rotation of axes it becomes ax- + hy^ + 6'^^= 1, we 

 shall have (from the usual properties of invariants) 



a4-6 + c=r(l+ 7i^ + ^-), \ 



ah + bc + ca = ^ P^- (1 + m^ + ^-), I 



abc = ; j 



so that the quadric is evidently a cylinder, and the direction of its 

 axis is the line x = y^ = z^". 



Suppose that c = ; then the area of a right section of the 

 cylinder will be 



-rr/^/iah) = |^/V3 (1 + w^ + ^^). 



But the angle between the normals to the right section and the 

 F-plane is the same as the angle between the two lines 



and x = y/'^ = zl'^^; 



i.e., is cos-i {3^7(1 + n^ + ^-)} : 



