412 PROF. A. C. DIXON ON STURM-LIOUVILLE HARMONIC EXPANSIONS. 



2. The equations (2) are somewhat simplified if we take (<7/&) 1/2 dx as independent 

 variable, and with a change of notation they become 



* < />=\p(J.,-, <1> = j (<r-\lp)(t><l.i' ....... (:{) 



Here a-, p are known functions of .r, p being positive, and it is assumed that the 



integrals of |<rj, p, - exist; X is a parameter independent of .<, and our first object 

 P 



will be to find an approximate solution when | X | is great, but A not necessarily real. 



3. Values of 0, 4> satisfying (3) and such that, when x = a, <j> = 0, and 4> = 1 will 

 be denoted by (x, a), <I> (x, a) ; if, when JS = a, = 1, and $ = they will be 

 denoted by \[s(x, a), ^ (x, a). If $ = A, $ = B, when x = a, then the equations (3) 

 become 



<f> = A+ 1 p<b dx, * = B+ [ (<T-\jp) <f> dx, 



Jo .0 



and have, according to the known theory of integral equations, a unique solution 

 which must be 



<j> = At//- (-f , ) 



<!' = A* (.r,) + B* (x, ). 

 Thus, it follows that 



<(> (.r, l>) = $ (a, b) ^ (x, a) + * (a, h) <t> (x, a), 

 ^ (.r, 6) = ^ (a, 6) >/r (, a) + * (a, &) tf> (x, ), 

 0> (;r, b) = $ (a, b) * (x, a) + 0> (a, &) * (x, a), 

 * (., />) = V' (, &) * (.*', a) + ^ (a, &) * (.r, a). 

 Other important relations are 



^. (x, b) <f> (a, c)- '/> (.r, t-) v> (, &) = ^ (, a) ^ (?>, c), 



V' (', &) V' (, <')-f (', <0 V' (, '>) = </> (x, a) * (<', ft), 

 and so on. 



4. If a second pair of equations of the type (3) is taken where /, a-', A', <j>', <' take 

 the places of p, cr, X, $, <f>, we have 



*'}<** ....... (4) 



by the formula for integration by parts, and similarly 



'}<&, ....... (4') 



