- 578 
[878 
878. 
NOTE ON THE ANHARMONIC RATIO EQUATION. 
[From the Messenger of Mathematics, vol. xvii. (1888), pp. 95, 96.] 
Given any four quantities a, ¡3, 7, 8, if 6 be one of the values of the anharmonic 
ratio, the other values are 
0» + 
1 
1 + 6 ’ 
e 
1 + o ’ 
i + e 
e ; 
and hence the equation having these six roots is 
(x-e)<^+ \ +6){x + (* + j-^) (U^r)“ 0 ; 
or, multiplying out, the equation, as is well known, takes the form 
(^2 + ^ + 1)3 _ ^ + 1)2 = 0. 
But to effect the multiplication in the easiest manner we may proceed as 
follows: writing 
a, h, c = (a — 8) (/3 - 7), (/3 - 8) (7 - a), (7 - 8) (a- /3), 
so that a + 6 + c = 0, the equation is 
The product of the first pair of factors is 
^ + 1 ~ (c + b) X ’ = ( xJrl )‘ 1 ~y c x ’