Mr. J. Cockle on a Theory of Transcendental Roots. 147 

 and, by the elimination of a;, 



^- i )( 3 3 3+9 £ +3 =°- 



Multiplying this result into vl~« 2 , and making 



-^^■i=?> 



it becomes 



f 3 -3f + 2\/l-« 2 =0, 



which is of the same form as the given cubic. Hence, since 



sc = (j)a, 

 therefore 



and 



dx _ fys/l—a 9, 

 da~" &Sl — a*' 



of which a corrected integral is 



n . /27T+ sin -1 «\ 

 # = 2sm( 5 ), 



which expresses the mean root of the cubic. And if we call 



# 1 = 2sml — ^ — ) =9i«, 



. /27r+sm~ 1 a\ , 

 %o=2 — )~9<2. a > 



F «=* !Hfl \ 3 



# 3 = 2 sin y g J = cj> 3 a, 



we have 



~da~~~~~ Zs/\=d? 



dfycfi _ <£ 2 v 1 — a 2 

 ~da~"~~~ 3^1"^?' 



d(j> 3 a _ ^x/l — « 2 . 

 and the mean root alone satisfies all the conditions of the pro- 



