244 Mr Blythe, On the Construction of a model [Oct. 29, 



On this drawing-board make a figure similar to I., it will not 

 be necessary to draw the lines 1, 11, 18, but 4, 9, 13 should be 

 marked, and the points along the edges at which all lines cut 

 them. 



Consider the plane of the drawing-board as being a certain 

 distance, about five or six inches at least, below the plane of I. 



It will be easy by the help of the drawings to determine the 

 heights of any points on the straight lines, either on the edges of 

 the figure, or at any other convenient points within it. 



When choosing these points the position of the projection of 

 the line on the drawing-board may be obtained by fixing two 

 drawing pins where it meets the edges, and joining them by a 

 silk thread. 



A small model may be made on a much smaller board, the 

 lines being passed through the eyes of needles fixed into corks at 

 the requisite heights, the ends being tied down to the sides of 

 the boards. In a larger model however it will be found more 

 convenient, when the points have been marked down on the 

 drawing-board, and the height of each indicated, to get blocks of 

 wood in which uprights of the required height have been fixed. 

 An upright may be of metal, or wood with a screw-eye at the top 

 of it. If the measured height be not quite accurate these blocks 

 can be moved backwards or forwards till the string is exactly in 

 position. The wire representing the line, passing through the 

 screw-eyes, is firmly fixed at its ends to the edges of the board, and 

 the blocks can be fixed by means of hinges, or pieces of wood 

 glued at the side. 



The lines in the figures II., III., IV., should be first attached 

 to uprights carefully measured and fixed in position, for three 

 being in each vertical plane adjustment is more difficult. 



It will be found that some lines require to be attached to the 

 board at one end, the other only requiring an upright support. 



The positions of the straight lines may be found by analysis 

 by taking the equations given by Cayley of the forty-five planes : 

 express the condition that the plane (9, 20, 25) that is the plane 

 Imn (p — /3)2 + [Im (p — a) + 2n] to = is at right angles to w = 0, 

 and then take 1 — — S, m= — 2, n = 2-^^. Since a and /3 are known 

 in terms of I, m, n we can find the value of p, and k, and all 

 constants are known. Use areal coordinates. 



\See Collected Math. Papers by Cayley, Vol. i. Number 76.] 



