Definite Integrals, with Physical Applications. 375 



16. To find a formula, which shall always represent the least 

 of two quantities a and /3, abstracting from the sign. 



It has been shewn, in the paper on the Resolution of Equa- 

 tions*, that if <j>(x) be any function of x involving only positive 



and integer powers of x, the coefficient of - in — h.l. ^— i i s the 



same root of the equation 0(#) = O as that given by Lagrange's 

 theorem, that is, the least. Hence it follows that if we seek the 



ec • e 1 ■ -l i {x + a).(x + (i) . . . / afZ+X°\ ., .„ 



coefficient of - in h.l. '— '- J - or m h.l. 1+ — j- — -ftA , it will 



xx \ x(a + fi)J 



represent the least of the quantities a or /3. Expanding, there- 

 fore, this logarithm, and taking the coefficient of - in each term, 

 we get the equation, 



r r , , . ,\a a/3 1.1 (a/3) 5 1.1.3 (a/3) 1 



J he least of \ . = — ^ + -p—r . ^' , + , . a ^— 5 + &c. 



1/3 « + /3 2.4 / a + /3 \ 3 2.4.6 <> + P \ 



17. Put - for a and y. for /3; we find thus: 



a fi 



ti i t c « 1 LI "ft 1-1-3 («/3) ? , B 



2 I 



representing this series by S, it follows that 

 d"- l .S 1 



when a >/3, 



(-1)"-'.1.2 (W-I).da"- 1 a"* 



and =0 when a</3, 

 that is, we have a formula which represents or — according 



a 



as a is < or > /3. 



* C'amb. Trans. Vol. IV. Part I. 



