SURFACES IN HYPERSPACE. 335 



The condition on the surface due to the vanishing of these invariants 

 is as follows: 



(1) h- = 0, minimum surfaces (§36), 



(2) G = 0, developable surfaces (§41), 



(3) [fJix5]2 — <|>2s/a" = 0, surfaces with what Levi calls axial points, 

 viz., three dimensional surfaces or surfaces formed by the tan- 

 gents to a twisted curve (§43). 



(4) [|JLx8]2 = 0, surfaces with perpendicular (Segre) characteristics — 

 the indicatrix reduces to a linear segment — the simplest gen- 

 eralization of ordinary surfaces (§43). 



(5) [jJi'xSxhj^ = 0, surfaces possessing (Segre) characteristics — sur- 

 faces with what Levi calls planar points (§43). 



Conditions (1), (3), (4) imply (5); condition (3) implies (4). We 

 have already discussed minimum surfaces briefly; we shall take up 

 the other types in some detail in later sections. 



