Definite Integrals, with Physical Applications. 381 



Denote by <p„ <p>, <fa, &c. the preceding functions, and let the 



a- ■ <. 1 . , , (x + a)(x + 3)(x + y)... , „ , 



coefticient of - in h.l.- '- MM r ' be found, and be re- 



X x 



presented by «S', then shall 



. dS . dS dS _ 



be the required formula. 



For when a is the least of the quantities «, /3, 7 &c. then 

 S simply represents a, and therefore 



dS , rftf „ rf<S* 



and the above formula reduces itself to <£>,. 



Similarly, when /3 is least, it becomes = <£ 2 ; and so on. 



22. To rind a formula, representing a discontinuous function 

 <p, which assumes the successive values <p t , <p._, <p 3 , &c. according 

 as a variable quantity %, commencing with a value = a, flows 

 through the successively increasing magnitudes /?, 7, S, &c. 



Denote bv S («, *) the coefficient of - in h. 1. (* + «)■(* + «) 



£</3,.) h.l. **?)■(»+») 



&c. &c. 



and suppose <£ =./, rf g +^- Jo +■/« r // + &c * 



Now when . > . I ^ ££(**) =1 Mjg,,) ' rfj( 7 , g) 



< /3J f/a r//i r/7 



and, by hypothesis, (/> is then = <^,. 



,'i C2 



