PROF. A. C. 1HXOX MX VITRM UOUVILLE HARMONIC EXPANSION^ 429 



which reduces to 



if ft is a riMit of the equal inn 



(15) 

 (19) 



When GH EK L 1 is ptwtitive, fti, fa, the two roots of ( I!)), ;ire conjugate complex 

 quantities, and they may still be considered so when GH EK = L* and 0,, 0, are 

 rrnl and equal. Let ,, l>e the two corresponding solutions of the fundamental 

 equations (3) ; these are ulso conjugate. Let coefficients 



anil 



l>e so determined that '<, l> n are conjugate for all values of n, and that 



. (20) 



is the least possible. Here the different values of X satisfying the equation (15) and 

 the corresponding values of ., are distinguished by suffixes. Thus the two factors 

 in the subject of integration are conjugate imaginaries and the integral is essentially 

 jxjsitive. 



From (5), 4, we have 



'() 3.() 



/> 



H,(o) 



\rli EK1 /. 



I- from (18) 



Similarly, 



= 0, since 0,0, = GH-EK. 

 r l I 



for any unequal suffixes , n. 



Hence it readily follows that (20) is a minimum when 



. = f - /() > Or) dx -5- | - & (-r) I, (x) dx, 



fi i .-' I 



1> \ - f(r\t latltLr -=-1 - f ( r\ iv\ilv 

 " J V J / V* 1 '/ u- * V* 1 / 'J* V 1 * / ' i 



Jo n Jo p 



whilf tin- gfiirral term in the expansion of./'(x) ( 19) is 



