206 MR. ROBERT RAWSON ON THE CUBIC INTEGRAL. 



6. Transformation of the cubic integral (8) by the re- 

 lation 



2 = cosm0, (9) 



where m is any positive whole number. 



From (8), dz=— m sin Odd = —m sj \—z^dQ ; then (8) 

 becomes 



► I _,/2j3-o-6\ 



/ 2 \ dd . . 



uz=m\/ ——f 1 — / — (10) 



\ / H n- cos mu 



V a + o — 2c 



1 _i/la— o— 6\ 



I — cos ^l -^ — ) 



m \ a—o / 



Let Mj, represent the integral (10) between the limits 

 a = a and ^ = b, then 



"=r 



dx 



\/{a—x){b — oc) [c—x) 





}' (lO 



, cos mv 



+ b — 2c 



By comparing (6) and (11) we obtain : — 



C dx C dx , . 



I — =1 — r =^-(12) 



Jj ^{a—x){b—x){c—x) J_^ \/{a-x){b—x){c—x) 



This result is neat^ and gives an answer to question 5508 

 of the 'Educational Times' for December 1877. 



7. Since 2a< 2a, then 2a,~ a—b < a — b ; therefore2a — 

 a — b upon a — b and 2^ — a — b upon a — b are always proper 

 fractions, whatever may be the values of the real positive 

 roots a, b, c. 



The limits of both (5) and (10), integrals which repre- 

 sent the cases in i and 2 in art. 2 respectively, are angles 

 0f and 6^, which lie between zero and tt. 



