OF HEAT IN ELLIPSOIDS OF REVOLUTION. 
133 
11 . We now return to the consideration of equations (22), confining ourselves in the 
first instance to the first system which, for the sake of convenience we shall rewrite, 
p iai =-( Ko -k)a 0 
1 
p. 2 ct. 2 = -{/q— k)a x — « 0 
Ps a s = ~{ K -2—fy a . 2 —cq 
Pr+I^r+I — ( K r k)ct y Ci r _ j, 
6 
in which e=X 3 c 2 , 
2 n~ + 2 n — 2m? — 1 
*' = l2^-l)(2n + 37 
, . „, (?i 2 — m?){n— l 2 — on?) , ^ 
£+)i („ + l), n=m+2r. 
We are to endeavour to discover the nature of the convergence of the series 
a 0 , a u a. 2 , . . . and the nature and position of the roots of the equation a x =0. A 
similar system of equations has been discussed by Heine in his ‘ Handbuch cler 
Kugelfunctionen,’ second ed., p. 406, and the following investigations are mainly 
modelled on the principles which he has used. 
( 1 ) Treating the constant a 0 throughout as positive, we observe that the series 
a 0 ctj . . . a r are all positive when k— — oo , and alternatively positive and negative 
when k= + oo . Moreover, as in Sturm’s functions, no change of sign is lost or gained 
by the passage of any of the intermediate members of the series through zero; and 
since the whole series gain r changes of sign as k passes from — co to + oo , it follows 
that all the r roots of a r — 0 are real. 
( 2 ) We shall now show that all the r roots of a,.= 0 lie below k,- ; but, previous to 
doing so, we must inquire more closely into the values of p r and -(/c ,. +1 — k,). 
(n 2 — m 2 ) (n — 1 2 —to 2 ) 
16fa 3 — i)(n — l 2 -f) ; 
and U r=a> p,—TQ. 
hence p,- lies below yg- (neglecting the case of m— 0) 
(a) p,.= 
(&) 
-(*>+ \ — K>) 
4% + 6 4m~ — 1 
A 2 c 2 + ' (2»-l)(2»+3)(2»+7) * 
(41) 
and, as the present discussion turns upon having this quantity greater than 1 + 
we must assure ourselves that this is the case. Now, since Xc is not always necessarily 
small, and n need not be large, this expression is not always > l+yg-; but by choosing 
r and therefore n large enough we can ensure that this is the case whatever value Xc 
may have, and the present proof will commence with such values of r as certainly give 
