Definite Integrals, with Physical Applications. m7 



therefore h. l~= h. I. (l + ^-k ) , h , /, , «r,> - b,\ B 



Q [ 1 + x + bJ +hA - 1 1 + .^p^J +&c.=4(c). 



Hence by Ex. 3, 



f ® = thht)- {(i "' +t '' + '-' ta -)-(t t >+t t *+...t>.)\. 



For other instances of Logarithmic functions, we refer to 

 Note (A) at the end of this paper. 



10. Another form of 4(c) to be considered, as frequently oc- 

 cnrnng, is that of fractions of which the numerator and deno- 

 minator are composed of a variable number of factors the 

 relation between 4(c) and 4(c + l) wi „ enable us to ^^ 

 such a function into a series of simple fractions; we may then 

 recur from 0(c) tof(() as before. 



Ex. 6. Given 4(c). _Ai^i^_ f - , 



Y ' 3.5.7 (2c+l)* to nnd /W« 



By the nature of this function we have 



(2c+2).4(c)=*(2c + 3).4(c + l). 

 Substitute for 4(c) the series 



A' B- c 



— ; — + 1 8lo ■ 



c+m c + ffl + 1 ^ c + »+2 ' 



the equation becomes by actual division, 



g^a^g) 2J.(l-»-l) 2C.(l-„_ 2 ) 



x + n x + n + l x + fi + o & c - 



_2^'(|-»-l) 22?.(§-» T#2 ) 



c+-» + l x + n + 2 



-&c. 



