134 
PROFESSOR C. RIVEN OR THE CONDUCTION 
k,_o)> 1+T6- When this is true it follows, a fortiori, that - (k,_ — k\._ 3 ), 
— K r _f) . . . -(k, — K,_fj, -(*>■+! —Kr) ■ ■ • are all greater than yf. 
6 6 6 
Let us also suppose, for the sake of clearness, that r is even; then, if we substitute 
ic=K r _ 1 in the equations, cq, cq, a. 2 , . . . cq_ 3 are alternately positive and negative and 
each is less than -fa of the one after it; we have, in fact, 
P\ a \— ~( K r-i K o) a o 
P2 C k = —~( K r-l — K l) a l — <X 0> &C - 5 
and since when ]c=k,-_ 1 , p,a r — —oq_ 3 , cq will also be negative. But, in the same 
manner, when we substitute k=K, or any greater quantity, the series cq cq ... a,, are 
alternately positive and negative, and remain so as k changes from k,- to + 20 • We 
infer, therefore, that all the roots of a r =0 lie below k,- and that one root lies between 
/c,_ 1 and K r . 
(c) Moreover, only one root of a, — 0 lies between k,_ x and k y . For when k = k,_ 
the series cq . . . a,._ 1 have r — 1 changes of sign and retain these ever afterwards as k 
increases; therefore the subsequent changes of k can introduce only one more change 
of sign into the series a 0 , cq . . . a r . 
(cl) When k—p, one of the roots of cq= 0, the expressions -(k,- +1 —p),~(k, +z — p). . . 
6 6 
are all positive and greater than 1 + yg-; and, for this value of k,p r+2 a r+2 =-(K r+1 —p)a r+1 , 
Pr+s a r+B = ~( K r+z — p) a r+ 2 — a r+\ • • •', thus a r+1 , cq +2 . . . have all the same signs and each 
is less than -yg- of the one after it, and these signs are opposite to that of a r _ v But 
as k increases from one root p to the next p of cq= 0, « ; ._ 1 must have undergone one 
change of sign; hence cq +1 , cq +3 . . . must have each undergone one change of sign. 
In other words, each of the equations cq +1 =0, cq +3 = 0 . . . has one root between each 
pair of a, — 0. 
(e) It is clear, therefore, that each of the functions cq, cq . . . vanishes once- for values 
of k lying in the intervals between — oo , k 0 , /q . . . and but once. Let us there¬ 
fore conceive these lengths cut off on the axis of k, and construct the curves a,+i—fi(k), 
a r =f 2 (k), cq_i =f s (k) . . . When «,== 0, a r+x — — lGc/,._ 1 ultimately when r becomes 
6 
very great; and when cq +1 = 0, a,, and a,_ l have like signs and a r — — —a r _ v a r there- 
fore becomes indefinitely small compared to and reference to equations (22) shows 
that then cq_, is indefinitely small compared to cq_ 2 . . . . The points, therefore, in 
which the curve a=f. : (k) cuts the axis converge to fixed points as r becomes infinitely 
