MR. ROBKRT RAWSON ON THE CUBIC INTEGRAL, 201 



XXVI. On the Cubic Integral u = \ — 



J^ \l {a—x){h—x){c—xY 



By Robert Rawson^ Esq., Assoc. I. N. A., Hon. 

 Memb. Manchester Literary and Philosophical Soci- 

 ety, Memb. of the London Math. Society. 



Read January 22nd, 1878. 



I . As far as I know, the cubic integral has received but 

 little attention from mathematicians ; the reason for this 

 may be that it has been regarded by them as a particular 

 case of the quartic integral. 



This, no doubt, is, to a certain extent, true. Still, the 

 quartic integral is readily reduced to the cubic (see Art. 13), 

 and there are some advantages in considering the cubic 

 integral first, in its natural order, especially so in the re- 

 duction of it to an elementary integral of a fractional mo- 

 dulus n and amplitude d. 



The quartic integral here alluded to is usually called 

 elliptic integral. I cannot hope, however, to induce ma- 

 thematicians to abandon this unnatural terminology, viz. 

 elliptic integrals, for the more natural one of quartic inte- 

 grals, as suggested by Professor Cayley (see Salmon^s 

 ' Higher Algebra,^ page 83). 



In this paper I have not entered into the great question 

 of the comparison of cubic integrals ; my effort here has 

 been of a more limited kind, viz. to reduce the cubic inte- 

 gral to the elementary integral, 



Jo Vl 



dd 



+ n cos md 



