204) MR. ROBERT RAWSON ON THE CUBIC INTEGRAL. 



a + b — 2c ( (a — b)z ") ^^^/^ \ 



i+z L a + b — 2c) i^-z 



The limits become as follows : — 



, « + a — 2C 

 When x = », then z= ; 



a + /3 — 2c 



Substitute these values in the cubic integral u, and it 

 becomes 



^a+a — ze 

 a— a 



"~" V a + 6 — 2c\ / ^ / {a — b)z \ ' 



a+g-2c V ^^ '^''v a + b — 2c) 



The value of v/« + 6~2cis real, because a + Z> — 2c is 

 positive. 



The fraction a — b upon a-]-b — 2c lies between i and 

 zero; it is zero when a = b, and i when h=c. At each of 

 these limits the cubic integral is soluble. 



4. Transformation of the cubic integral (3) by the re- 

 lation 



— ^' = coswi^, ^ (4) 



where m is any whole positive number. 



From (4), dz =m sin mdd6=m >J I— z^dO. 

 Substitute these values in (3), and it becomes 



I — i/ic— a— a\ 



-cos I ) 



m \ a— a ' 



u = m\/ ——: 1 — / , =^' • (5) 



V a + b — 2c\ / a — b ^ 



\/ i-\ i cos mO 



V o + 6 - 2c 



—cos '( r-!- 1 



