58 Proceedings of the Boyal Society 
treated in the same way, and thus we form a series of equations— 
A = pf> 4- qfi + D 
B ~ pf + + E 
C — p s D 4 - ^ 3 E + F, Ac., 
in which p can never be zero, while q may he so. 
In order to compute, by help of these quotients, the approximate 
ratios of A, B, C, we may put A x , A 2 , A s , Ac.; B 1} B.„ B 3 , Ac.; C x , 
C a , C 3 , Ac., for the corresponding successive values, and then we 
obtain the equations— 
An -{- l = pn 1 A n + qn Aft _ i + Aft _ 2 , 
Bft-}-l = pn- j-1 B n 4* qn B ra _ 1 + B rt _ 2 , 
On-J-1 — pn + 1 Oft 4 ~ qn Oft—1 + Oft _ 2 5 
which indicate a very simple arrangement, best studied from an 
example. Thus, if the successive equations were— 
A = 2.B 4 - l.C + D 
B = 3.C 4- 2.D 4- E 
C = 2.D + O.E + F 
D = 3.E + l.F + G 
E = 2.F + 2.0 + H 
F = 3.0 + O.H + I 
O = 2.H 4- 1.1 + K 
H = 3.1 4- 2.K 4 - L, Ac. 
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, Cun lines below them. Unit being written as the first value 
of A under p v which in this case is 2, w r e multiply this by 2, and 
1 
1 
1 
1 
1 
1 
1 
1 
i 
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 
/ 
19 
H 
... 
... 
... 
1 
3 
8 
I 
... 
... 
... 
1 
2 
K 
... 
... 
1 
