162 Mr J. Larmor, On Possible Systems of Jointed [May 26, 



on a straight line, since the correspondence is of the first order or 

 linear. Thus to a generator corresponds a generator, and the points 

 of intersection of pairs of generators also correspond. Again, all 

 points corresponding to a given point lie on a curve which cuts the 

 system of surfaces normally, being in fact the curve of intersection 

 of two confocals of the other kinds. So that if we consider any 

 generator and the corresponding one on a consecutive surface, the 

 lines joining their extremities (where they meet generators of the 

 other system) are normal to the surface, and therefore to the 

 generators, and the generators are therefore of equal length 1 . The 

 condition necessary for deformation is thus satisfied, and the sur- 

 face, supposed made up of jointed rods, may be deformed without 

 straining into the consecutive confocal surface, and therefore by 

 successive steps, into any other confocal surface. 



We propose to investigate directly the cause of this want of 

 stiffness, and to determine the number of degrees of internal free- 

 dom possessed by other systems (which we shall prove to exist) 

 composed of three sets of rods connected by ball-joints, there being 

 three rods at each joint. 



In discussing the first problem, we may confine our attention 

 to three rods crossing three other rods: for we shall prove that 



Union 



Um 9 n 



h n h n i 



XjiUjCi "hnlJLnVc, 



X 3/ tt 3"3 



every other rod that crosses one set of them meets each rod that 

 crosses the other set, at a point in the rod which is unaltered 

 by the deformation. And for similar reasons, we shall only have 

 to consider in the second case the quasi-cubical framework formed 



1 Cf. H. J. S. Smith, Proc. Lond. Math. Soc. n. p. 244. 



