172 Mr. Horner oil the Solutions of the Function ^' x. [March> 



Pi'ovided r and p be real quantities, the case admits no other 

 periodical solution besides this. For r"— p" has no real binomial 

 divisors but r — o and r + p, and its trinomial divisors are all 

 of the form r- — 2 cos. ^ .r p + p", which in this case is neces- 

 sarily > 0. 



It is true, that since r and p, as the consideration of the hypo- 

 thesis r — f) = has proved, are always unequal, the following 

 conditions will obtain in the general case of real roots ; viz. if 

 we take r to be the greater root, and n infinite, we shall have 

 q^ = r constant, and thence 



/•cc^=l±(!l:zflf = Ji- (25) 



J '*' {r~b) + d.v r-b ' ' 



a constant quantity. To this form then, which must be called 

 oscillating rather than periodical, the condition of real roots ulti- 

 mately approaches. 



7. The uistance of a ball impelled through either focus of an 

 elliptical table, and continually reflected from the rim, affords 

 a practical exemplification of this ultimate tendency to simple 

 oscillation ; for after an infinite number of reflexions, it will 

 vibrate cDutinually in the transverse axis. The analytical expres- 

 sion of the facts would lead to some unobserved properties of the 

 elHpse, if this were the place for examining into them. It is as 

 follows : let S, s, represent the focal sections of the transverse 

 axis, e the excentricity, and x the half difference between the 

 distances of any point of impact from the foci ; then the similar 

 half diffisrence at the Mth subsequent impact will be 



which continually approximates toward the constant value e. 

 (See L. D. ubi supra.) 



8. After these partial or questionable instances are dismissed, 

 it appears that imaginary values of r, p, are essential to the 

 periodic character. The conditional equations (19), therefore, 

 are, 



" Mil. S ^ sin. J \'^ COS. ^J ^ ' 



Assume S= — , and these conditions are satisfied, provided k be 



n 



an integer, and 2 U not divisible by n. 



The reason of this restriction is sufficiently obvious ; for, 

 putting 2k = m n, the formulae (27) become, when m is even, 



s. = n (cos. s mit . V B)""' = n (i A)""' 



expressions which, n being arbitrary, can only vanish in conse- 

 quence of the simultaneous non-existence of A and B, and .•. of 

 r and p, and in short of the entire hypothesis. When m is odd, 



