Definite Integrals, with Physical Applications. 377 



19. To find a formula, of which the value is ra» when 



a — llfi 



«>/3, and . when ji>a; k being any quantity less than unity. 

 The order, in point of magnitude, of the quantities r— . and 



a — lip 



o_i » i >s tne same as the order of - and -5. For suppose a<ji 

 then a + ha<p + kfl; and, subtracting A(a+/3) from both sides, we 

 have a — hp t <p'-ka, attributing to h any value between and -=; 



again since A</3 we also have fHi + alt<ji + a; subtract ah + a from 

 both sides, .\ jilt — a < ji — alt, attributing to h now any value be- 

 tween -5 and 1 ; whence it is evident that for any value of h from 



to 1, we always have a — jih (abstracting from its sign)</3— ah, 



1 1 



when a<fi: and therefore if- be >, =, or < 3 accordingly shall 



a ji 



rr be >, =, or < 



a-jih "* ' ' "* ji-ah' 



Now by Art. 17, we have 

 the least of 



a-(ih 1 



1 (a + fi)(l-/l) 



[ji-all 

 1 1 Plafi.il+Jiy-ll.ia + jiy] 1.3 1 2 i \a(i.(l+ll)--h.(a+jiYY' . 



+ 2"2' (a + j8)'.(l-A) s + 2.4"3 - (« +/3) 5 .(1 -A) 5 ' 



which is obtained by putting a-fth for a, and ji — ah for /3 in the 

 article referred to. 



This series may be easily arranged in the more convenient 

 form, 



