MR. ROBERT RAWSON ON THE CUBIC INTEGRAL. 203 



necessary to trace by the usual rectangular-coordinate 

 method, the curve whose equation is 



V [a—x) [o—x) [c—x) 



Let Ox, Oy (fig. i) be rectangular coordinates, origin at O. 



When x=o, then OR Vabc^=i. The negative branch 

 of the curve extends to infinity with a continually dimi- 

 nishing value of the positive ordinate. The positive branch 

 extends only as far as A, where OA = a. All values of the 

 ordinate y beyond this point are impossible. The ordinate 

 y is impossible also between B and C, where OB = i and 

 OC=c, since the denominator of (i) is then negative. 



When a?=a=OA, then y=cx) . 



<r = Z»=OB, „ y=oo . 



x=c=OC, ,, y=oo . 



x<a and >b, » V is real and positive. 



x<c and >c, ,, y is impossible. 



x<c, „ y is real and positive. 



In this geometrical representation of the cubic integral 

 there are two cases to consider, viz. : — 



1. When the limits u and /S are both less than c. 



2. When the limits a and /3 lie between A and B. 



In case i the cubic integral is correctly represented by 

 the area HDEK, where OE = a and 0D = /3. 



In case 2 the cubic integral is geometrically expressed 

 by the area LFGM, where OG = u and 0F = /3. 



3. Transformation of the cubic integral u into another 

 cubic integral by the relation 



2c — a + a2 lia — c) 

 x=: =a ^— — . ... (2) 



1+^ l+Z ^ ^ 



From (2), dx=i-^ rl^z, a—x—-~—^ — -, 



