564 BELL SYSTEM TECHNICAL JOURNAL 



where N = 6 -\- iv, 4>u{x) = \^y{x)2'~"'4>{x). Introducing the parameter 

 w does not constitute an essential modification of the procedure 

 given by Laplace, but we shall find that w plays an important part 

 when we come to applications of the expansion presented in this 

 paper. 



In what follows it is assumed that y{x) is a positive monotonically 

 increasing or decreasing function in the range .ri, .r2; the extension to 

 a range of integration divisible into subranges within which y(x) is 

 monotonia will be obvious. 



Let X be that one of the two limits .ri, x^ for which 3'(.v) has its 

 greatest value; set y{X) = Y. Assume with Laplace that 



(2) [log Y - log t(x)]^ = (x - X)g{x, X), 



n being a positive number or zero and g a function of which {x — X) 

 is not a factor. Set 



(3) y = Ye-"-^', 



(4) c}>^..{x) (dx/dt) = ^o + ^iY-j + ^22i + ^3 3j+---. 



These two equations give (certain well-known conditions being 

 fulfilled) 



i_e_ai^ 



Finally set iV/"+^ = T and we obtain the modified LapJacian expansion 



u + 1' 



Ti = A^log Y - log 3'(.vi)]. = if X = .n; 

 T2 = Nllog Y - log y{xi)2, =0 if X = .T2; 



f 



Fills, T') = ^ 



T"-^e-TdT 



Tills) 

 V = Ti or T2. 



