58 Proceedings of the Royal Society 
treated in the same way, and thus we form a series of equations — 
A = 4- 4- D 
B = p . 2 G 4- q. 2 J) 4- E 
C = p.f> -f 2 3 E 4- F, &c., 
in which p can never he zero, while q may be so. 
In order to compute, by help of these quotients, the approximate 
ratios of A, B, C, we may put A v A 2 , A 3 , &c. ; B w B 2 , B 3 , &c. ; C 1} 
C 2 , C 3 , &c., for the corresponding successive values, and then we 
obtain the equations — 
An -j - 1 = pn-\-l A n 4" qn A n _i + A n _ 2 , 
B n 4-1 = Pn -f i Fft 4- q n B^ — i 4 ~ B« — 2 j 
Qn -f- 1 ~ pn 4- 1 Qn 4 * qn Qn — 1 4 ~ Qn — 2 > 
which indicate a very simple arrangement, best studied from an 
example. Thus, if the successive equations were — 
A = 2.B 4- 1.0 4- D 
B = 3.C 4- 2.D 4- E 
C = 2.D 4- O.E 4- F 
D = 3.E 4- l.F 4- G 
E = 2.F 4- 2.G- 4- H 
F = 3.G- 4- O.H 4- I 
G = 2.H + 1.1 4- K 
H = 3.1 4- 2.K 4- L, &c. 
we should write the values of p, q , 1 in horizontal lines as in the 
accompanying scheme ; and the successive approximate values of 
A, B, 0 in lines below them. Unit being written as the first value 
of A under p v which in this case is 2, we multiply this by 2, and 
1 
1 
1 
1 
1 
1 
1 
1 
1 
2 
1 
2 
0 
1 
2 
0 
1 
2 
0 
P 
2 
3 
2 
3 
2 
3 
2 
3 
2 
A 
1 
2 
7 
19 
59 
144 
569 
1197 
4304 
11571 
B 
1 
3 
8 
25 
61 
241 
507 
1823 
4901 
C 
1 
2 
6 
15 
59 
124 
446 
1199 
D 
1 
3 
7 
28 
59 
212 
570 
E 
1 
2 
8 
17 
61 
164 
F 
1 
3 
6 
22 
59 
G 
1 
2 
7 
19 
II 
1 
3 
8 
I 
1 
2 
K 
1 
