1884.] Wickerwork, and their Degrees of Internal Freedom. 165 

 = aj) 2 c 3 . afy a - a&c, . afy v 



A' = h c i &Yi « + «i 2 - « 2 2 - O 



+ aA« 1 /3 3 (^+7 3 2 -c I 2 -7 1 2 ). 

 Proceeding in a similar manner we find a like expression for 



A A A 



cos 6 2 a a . Therefore, since cos b./* 2 = — cos a 2 a 2 we obtain a relation 

 of the twelfth degree between the 12 segments: and we may find 

 two other similar ones in the same way. It is to be noticed that 

 the diagram is not really symmetrical, so that we cannot proceed 

 from one expression to another by simple permutation of the 

 symbols. 



Having thus independently established the existence of these 

 four relations, we establish at the same time the flexibility of the 

 system of six rods. Now every line that crosses three of the rods 

 meets every line that crosses the other tbree. For, if we denote 

 the two systems of rods for an instant by 123... 1'2'3'... the planes 

 through 1' and 1234 cut all lines in a constant anharmonic ratio, 

 therefore 123 are each divided in the same anharmonic ratio by 

 1'2'3'4'. Now consider 4, which is drawn across 1'2'3' : the plane 

 14' with the lines 1'2'3' divides it in the same anharmonic ratio as 

 1, 2 or 3 is divided by them : so does the plane 24' with the lines 

 1'2'3': therefore the planes 14', 24' are met by 4 in a common 

 point, or, in other words, 4 meets 4'. Further, the point in which 

 each of these lines crosses another is unaltered by deformation : 

 for the relations already established are sufficient to determine 

 definitely the segments of these lines in terms of the segments of 

 the six rods : we can therefore replace the lines by jointed rods. 



In the case of the paraboloidal system, in which all the rods of 

 the same series are divided similarly, we have relations of remark- 

 able simplicity. For the orthogonal projection on any plane 

 consists of two series of parallel lines, and the segments of each set 

 of rods are proportional to their projections. By considering the 

 projections on two different planes, the above results follow 

 immediately. 



Let us consider now the quasi-cubical system of jointed rods. 

 In the first place, such a system is abundantly possible ; for 

 assuming the 9 rods connecting the three layers which lie the same 

 way, and denoting them by the 9 digits, from any point on 1 draw 

 the line which intersects 2 and 3, from the point in which it meets 

 3 draw the line which intersects 6 and 9, from the point in which 

 it meets 9 draw the line which intersects 8 and 7, and from the 

 point in which it meets 7 draw the line which intersects 4 and 1, 

 as in the diagram. The last line must meet 1 in the point from 



