APPENDIX. 71 



In order to obtain corresponding expressions for pseudo- 

 Bpherical space, let R and t be imaginary; that is, ^=Ii, 

 and t=tl Equation 6 gives then 



from which, eliminating the imaginary form, we get 

 , =l f log. nat. |5 



Here S Q has real values only as long as r=R; for r=1J, the 

 distance S Q in pseudospherical space is infinite. The image 

 in plane space is, on the contrary, contained in the sphere of 

 radius JR, and every point of this sphere forms only one 

 point of the infinite pseudospherical space. The extension 

 in the direction of r is 



ds Q= W 

 dc $ 2 -i 



For linear elements, on the contrary, whose direction is at 

 right angles to r, and for which t is unchanged, we have in 

 both cases 



