264 Mr. G. H. Darwin. [Mar. 18, 



Thus the partial fractions corresponding to the roots a and b are 

 11 11 



4a 1 a3 — a 4b lt ^— b 



(15). 



If the pair of fractions corresponding to the roots <x + (3i be formed 

 and added together, we find 



1 -g 1 (a?-q)+/? 3 



O-<0 3 +/3 2 ] 



The sum of (15) and (16) is equal to -j 



(16). 



aj 4 -Aaj 3 +l 



and 



f = -1 log(» co a)- — log(aj co b) - a i— JogTfiB -a) 2 +/3 2 ! 



+ — — J?— — arc tan -ZL? (17), 



Substituting in the first of (14) we have 



h—x 



^ajsaxexp. — arc tan 



(# CO b) 8bl [ (# — a) 2 + /3 3 ] SlaS+M 



(18), 



where J. is a constant to be determined by the value of i, which 

 corresponds with a particular value of x. 



From the third of (14) we see that by omitting the factor 

 xj(h—x) from the above, we obtain the expression for j. 



x°(x—h) 



To find the expression for e we have to integrate . , ' . 



x\—h% 6 + \ 



Now 

 and therefore 



[ x*(x-h)dx _ ±l (x*—hiS + l) - l/i xHx 



The integral remaining on the right hand has been already deter- 

 mined in (17). Then substituting in the second of (14), we have 



l)i 



(x co a) exp. f — — arc tan - — - 1 



(«C0b)8b 1 [(x — *) 3 + /3 3 ] S(a^+/3-) 



• (19)- 



where B is a constant to^be determined by the value of e, correspond- 

 ing to some particular value of x. 



