380 Mr. T. H. Blakesley on Logarithmic 



All the lines in this, which may be called the second cell 

 in the direction n, including those of the circumscribino- 

 circle are homologous in a ratio n : 1 with the corresponding 

 lines of the first cell, and the angular displacement relatively 

 to the first cell is a— j3. 



To the points FGr may now be connected a third cell, 

 constituted in a similar manner to the second, and so on 

 indefinitely. 



To AC may be connected a similar cell bearing to the first 

 cell the linear proportion of 1 : n and to this another, and 



so on indefinitely in the direction — . 



J n 



Such a line of cells may be called a logarithmic lazy- 

 tongs. 



It is clear that all such points as A, D, G lie upon an 

 equiangular spiral, the tangent of whose characteristic angle 



i * ^^ 

 is equal to , . 



x log n 



As all the cells of a series are similar, any motion involving 

 the increase or decrease of the angle between the bars of one 

 cell, will be accompanied by the same change in angle in all 

 the cells. 



The sides AD, CB may also have cells attached to them, 

 the same rules as before being observed, m will take the 

 place of n in the change of scale and y—8 the place of a— 

 in the change of direction. 



It is to be pointed out that if two cells in the m direction 

 be applied to BC and BF, the other adjacent points of the 

 two cells will coincide. In other words, the cell on BF may 

 be considered as derived from the first cell either by one 

 move in the n direction followed by another in the m direc- 

 tion, or by one in the m direction followed by a second in 

 the n direction. Whence it follows that the whole of a plane 

 surface may be occupied by a plenum of cells forming an 

 infinite lattice-work, in which, if the angle between the cross- 

 bars of any one cell is changed, an equal change takes place 

 in that between the cross-bars of any other cell. 



In the m direction the tangent of the angle characteristic 

 of the spiral through such points as A and C is equal to 



<y — 8 



log,??! 



There is a common pole for both the m and the n spirals. 



If for the sake of easy description we liken the plenum of 

 cells so obtained to a chessboard, the rooks' moves would 



