OF HEAT IN ELLIPSOIDS OF REVOLUTION. 
137 
To obtain a third approximation, we must substitute this value of k in the above 
equation (43). 
The third series of terms may be arranged 
g 4. f Pi | ib j | jb-a | P>+2 
0 '-l\ 0 O 01 0102 0 '— 80'-2 0 '-+ 10'-+2 
■ < .4 i 9 '-+i / Pi | Pa | | P< -i | P> +3 
06+1 \0001 0102 0 ,'— 20 r —1 0r+20/-+3 
while the coefficient of 0,. in the second series is 
.2/ Pi 1 P2 
rfx+ •••+j i h L + 
P»-+2 
, 0 o 0 i 0102 
0 r_ 20;-—1 06 + 104+2 
On putting, therefore, the value of 0,. given by (45) in these terms, we see that 
shall have 
P'~ I P'~+l \ | 4/ P'--lP'- I P'-+lP'-+2 
we 
j _ 2/ P'~ I P'~+l | 1 4[ ^I rf+irf+2 \ I 
9r ~ Wr-ir^r \0'- 2 0'v 1 _r 0' 2 , +a 0bJ" h • 
• (46) 
where, in 0 ",_ 1 , 0'b+i we have to substitute the value of A’ given by the second approxi¬ 
mation. This gives /j as far as e 5 . 
I proceed to find its value up to eh 
Having n=m+2r, n'=m+2r', we shall put 
D /V =«'(?f +1) -n(n- f-1) 
^ 2n ,2 + 2n'—2 / m 2 — l 2n 2 + 2n—2m 2 — l 
/V ~ (2ft'-l)(2ft' + 3) (2ft —l)(2?i + 3) ’ 
and, for a first approximation, obtain 
7 _ / , lX , 2 ft 2 + 2 ft— 2 m 2 -1 
4_»(»+X)+ (2b _ 1)(2k+3) e; 
for a second approximation, 
7 / ■ 7 \ * 2 ft 2 + 2 ft— 2 yft 2 — 1 
k=n(a+l)+ (2m _ 1(2m + 8) 
Pr 
+ 
I) ^ 1) 
P'’+l \ ,2 I (jPr$r-l,r , Pf+ 1 ^ 
€3 + 
r—\ t r 
D 2 , 
+i.<- 
and, as a third, 
2ft 2 + 2ft-2m 2 -1 
(2ft—1) (2ft+ 3) 
&=n(ft+l)-f 
(n^"+n ±L ) ^+(x# =Lr +^# ±ir ) e 3 ' 
■L'r+l.r/ \ 1J r—l,r 1 J T+I.r / 
4 T / P f I P r +1 V P>' I Pr+l \_ pr^~r—\,r _ Pz+i&'V+l/ 
1 \D^ T D w /lDVD»,J Dh-!,, Db +1 ,, 
P>-\P' P»+lP)-+2 1 ■ 
't> I ) 2 T ) 2 1 ) 
± -'r—% % r ±J r—i,r r+l,r- L/ /*+ 2 ,r J 
(47) 
(48) 
MDCCCLXXX. 
T 
