54 
NOTE ON MR JERRARD’s RESEARCHES ON THE EQUATION &C. [310 
Take Cardan’s solution of the cubic equation x 3 4- qx — r = 0, we have x = a + b, and 
thence Sab = — q, a 3 + b 3 = r; and to obtain the solution w'e write 
a 3 b 3 = — ~ , a 3 + b 3 = r. 
27 
But these two equations are not enough to precisely determine x, they lead to the 
9-valued function 
/ r /r 2 (f 
V 2 + V 4 + ^7 
+ 
tf 
27 : 
in order to precisely determine x, it is (as everybody knows) necessary to use the 
original equation ab = — ^ > . But seek for the solution as follows; viz. write x=ab(a + b), 
which gives 
3a 3 b 3 = — q, a 3 b 3 (a 3 + b 3 ) = r, 
or Avhat. is the same thing, 
a 3 b 3 — — |, a 3 + b s = — ; 
these equations give x = ab (a + b), where 
a = 
v 7 
/9 r 2 q 
+ A/ if + r 
b = 
9^ 0 
4g 2 3 ’ 
which is a 3-valued function only, ab in this case being not given. 
2, Stone Buildings, W.G., January 28, 1861.