212 MR. ROBERT RAWSON ON THE CUBIC INTEGRAL. 



.2 1 2 



v 



7C ' Vi — n 



I + n cos - 

 P 



The larger p is, the more accurate is formula (24) . 



12. The reduction of the quartic integral, usually called 

 elliptical integral, to the cubic integral is remarkably easy. 



To effect this, let the quartic integral be as follows : — 



* . (.0 



■r 



^{e-y){f-y){9-y){h-yy 



where €,f, g, h are in the order of magnitude. 

 Transform the integral (25) by the relation 



Fi'om (26), 



p + ex p — eq , ,. 



y=^^-\—=e+^^-—^ 26) 



ihz=-l — ^dx, e—y = -^ — ^, 



^ q^-x\ e — h ) 



Substitute these values in (25), observing the limits, 

 then 



-Vt^ 



eq—p 



ie-f){e-ff){e-h) 



>ag-p 



dx 



(27) 



\/fe-)(fE.MM-^l 



e-B 



This integral only requires that eq >p in order to obtain 

 its arithmetical value. 



