254 PROFESSOR A. G. GREENHILL ON THE 



a quartic for X, having a root X = 1, which was used for p. = 2n + 1 ; the remaining 

 cubic, putting X = 2B 1, becomes 



B 3 _ W + V--- 1 - l B - *- 1 + 1=6 (4) 



. A. 



or as in 20 with 



2c 

 so that B = reduces this to 



?/ + * 



2r 3 - (y + 1) c 2 + 2a;c a = (7), 



as in (3), 8 ; and with 



m 3 a (1 2m) a (2a l)m / R . 



x = , y = v 7 , ?/ + 1 = v (8). 



a m a. m a m 



c= - 7//a - B= 2a X== 2a + 1 (9). 



' - ' - 



w , 



2a - 1 ' 2a - 



We find that CLKBSCH'S k (CEELLK, 57, p. 105; L.M.S., 27, pp. 146, 185) is 



given by 



P = 4a - 1 (10), 



so that 



and 

 so that 



and in quadratic factors 



(15). 



Thus, as shown in the ' Bulletin de la Socie'te' Mathematique de France,' 29, the 

 algebraical spherical catenary for p. = 10, drawn stereoscopically by the late 

 Mr. T. I. DEWAR, can be represented, by taking k = y/f, h= ^x/V, ni 

 symmetrical form 



