62 
• dv 
then the coefficient of — in (14) is zero, and it becomes 
S-I=° < l6 > 
The general solution of (16) is well known to be 
y-(^ + (*-•)* (17) 
wher Ci, c 2 are arbitrary constants to be determined. Inte- 
grate (15) with respect to x, then, 
m 2 27 € 
Hence, equation (17) is a root of the cubic 
y 8 + ay + |-^e“* = 0 - (19) 
The values of c 1} c 2 are readily determined by substituting 
the value of y in (17) in (19), they are as follows : 
Ci— 
2 a 3 
27 
Substitute these values in (17), then 
(20) 
is a root of the cubic (19). 
(5) The values of c* £ _ in (19) can be determined by 
means of a quadratic, so as to satisfy the classical cubic 
y 3 + ay + b = 0 (21) 
This equation will coincide with (20) if 
(22) 
Then, e * = b -^/ b * + ^ 
a a 2a 3 -x 1 / 7 o 4a 3 h 
An ’ 27 e \2 + 2\f + "27 ". ) 
Therefore, 
The value of 2 / in (23) agrees with Cardan s formula. 
The process of obtaining the first and second differential 
