PROCEEDINGS OF SECTION H. 573 



of the circle, may be divided in a given ratio by geometrical construc- 

 tion. In the ordinary notation for elliptic integrals the length of the 

 arc OP 



where ^ is an angle such that s/sin 2Q = cos <^. Tables of elliptic 

 integrals are available, and with their help it thus becomes possible to 

 prepare a table of the lengths of arcs of the lemniscate for different 

 values of 0. Table I. has been computed in this way, and gives the 

 length of arc measured from and the increment of arc corresponding 

 to each increment of 30' in the angle 61 up to a limiting value of 30°. 

 The table is of perfectly general application, the tabulated values 

 having to be multiplied by the value of a for the particular lemniscate. 

 These are computed to five places of decimals, which is ample for the 

 purpose, the value of a being generally between one and two thousand 

 links. The increment of arc over 30'. may be taken as practically 

 equal to the chord, the maximum difference being less than 2 in the 

 last place of decimals. 



^li^ljl- To set out the Lenmiscafe. 



The first portion of the lemniscate curve is most conveniently 

 located by setting up the theodolite at (Fig. 1) and measuring off 

 the proper length of OP corresponding to a given value of d. Table 

 II. has been computed from the equation OP = av/ sin 29, and again 

 the tabulated values, have to be multiplied by a to obtain the corre- 

 sponding measure of OP. When the distance becomes too great 

 for this to be conveniently done, further points may be located by 

 means of Table I. and measuring along the curve. The tables are taken 

 to much greater values of 6 than necessary for ordinary transition 

 curves, so that they are applicable to the setting out of double lemnis- 

 ■cate curves or to any conceivable problem in which they might be 

 required. 



It appears to the writer that the lemniscate possesses the great 

 advantage over the cubic parabola that for all lengths of transition 

 curve we have a simple exact solution. Even when it is possible the 

 same degree of exactness can only be obtained with the cubic parabola 

 by much more complicated calculation. 



