PROP. A. C. DIXON ON 8TURM-LIOUVILLE HAKMOMC EXPANSIONS. 417 

 Since 



,r J 



= r V = -/ dor, 



'/./ . .' / 



the integral is a coiitinuottt function of x, it follows that the discontinuity in 

 l"g y > s equal to that in log /. Also / has no discontinuity, being equal t 



' / ./ 



1+ ["rVcfcs. 



. n 



Let the imaginary quant itirs conjugate to X, ;, ... In- denoted by X, f, ..., and let 

 if = rr, \/ \ = a + i/9, a being positive. Also write D for d/d.r. Then in each sub- 

 inter\;il 



Also, 



Hence, when JT = d, the value of (I)*+4f?) ir is U(a 2 +/3 : '), or 2JX', and that of its 

 derivative is 0. At any discontinuity when ; is changed to / (both are real) the 

 derivative is multiplied by /'// and (D'-i l/-f') ir itself consists of two positive terms of 

 which the first is unchanged while the second is multiplied by (f'/r) 1 . Thus the effect 

 on (I) :l +4/3 : ') ir is to multiply it by a quantity In-tween 1 and (r'/r) 3 . 



No\\. if in any interval a function // satisfies the diH'erential equation D a y = 4a ; //, 

 and if at the beginning of that interval // lies between A cosh 2a (jr a) and 

 B cosh 2a (./ ti), while I)// lies l)etween 2Aa siuh 2a(.r a) and 2B sinh 2a (x a) 

 where A, B, j- a are real and positive, then these same statements must hold 

 good throughout the interval. For suppose B > A, then Bcosh 2a(.r a) y and 

 y A cosh 'la. (.ra) are both functions satisfying the 



and at the beginning of the interval their values and derivatives are all positive; it 

 follows from the differential equation that the values and derivatives will increase, 

 and therefore be positive throughout the interval. 



In the first sub-interval (D'+Af?) u< must be 2|\|cosh 2a(x a) and its derivative 

 4a|X ! sinh 2a (x a). At entrance to any later sub-interval of the domain 

 ("./') ( < 6) this function and its derivative are multiplied by quantities of which 

 we know that each lies between 1 and T* where T is the ratio of increase in r. Thus 

 throughout all the sub-intervals 



(D*+ 4*) w = 2P | X | cosh 2<x (a a), (D s + 4^D) w = 2Qa | X | sinh 2a (ar-a) 



where P, Q are quantities lying between HIT* and liar 3 , if we use II, to denote the 

 VOL. CCXI. A. 3 H 



