242 



PROFESSOR A. G. GREENHILL ON THE 



The nine division values of the argument 2 r v, v = W3 should now be functions of 



I . ' 



an argument u + , and thence derivable from each other, being grouped in sets of 



three 



r, 8v, 7v 



v, 2 3 ?>, 2v 

 2v, 2*v, 2 7 r 

 , 2 5 ?>, 2 8 v 



or 



2v, 3-)', 5f 

 4v, Gv, 9r 



(11). 



In passing horizontally in these sets, the substitution connecting a = a (w) and 



(12). 



(13) 

 (14). 



7 / , iTtO \ 



b = a ftt + j is 



a?b z 2ab a 6 = 



The additional cubic transformation 



_ c 3 + 15c 2 +_57c 

 (3c + 19) 2 



leads to a multiplication relation of the 9' h order, connecting 



H = H (u) and c = H (4 u) 



The relation of the 12 th order between a and b is of the G th order in p = a + 6 and 

 ^ = a&, representing a C fi in the coordinates p and q. 



But so far the various transformations of y ]9 = 0, as given in L.M.S., 25, lead to a 

 C 7 of the fifth degree in each variable, and the reason of this is a mystery still. 



17. For /j, = 21, applying the trisection equation (8) 14, with the relations of 9 



x = z (1 z)~, y = 2(1 z) (1), 



Put 



+ 3:~ (1 - z)*(l - 82 + z~) t a ~ + 3^(1 - 2) 6 /, + (1 - z)' = (2). 



_ 



t = Z ~ ( - (3), 



- I) 3 = (4), 



(5), 

 (6), 



2 M; 

 + - 1) z 

 3-io s + - 



- ?* - 2it; 3 + 310 s8 + - 1 + 



,' = 



W = 8 -8 + 22 24 + 11+4 6 + + 1 



= (^2 w+ 1) [(ter - Stt- 2 + + I) 2 - (w 2 - w) (w 8 - 3w 2 +0+ 1) + (w 2 - w) 2 ] (7). 



