346 
Proceedings of Eoyal Society of Edinburgh. [sess. 
and, collecting those terms involving ^4, ^2, we get this new 
scheme, which makes it clear that the numerical coefficients in the 
development of the powers of ^4 + ^2 + 1 give the desired approxi- 
mations to the values of 1, ^/2 and of ^4. 
Those coefficients are readily found thus : — 
We write the group 
1 , 1 , 5 
0, 1, 4 
0, 1, 3 
and continue the progression by adding the antepenult to the triples 
of the last and of the last but one 
term for the succeedin 
g term. 
1 
1 
5 
19 
73 
281 
1081 
4159 
16001 
etc. 
0 
1 
4 
15 
58 
223 
858 
3301 
12700 
etc. 
0 
1 
3 
12 
46 
177 
681 
2620 
10800 
etc. 
If we follow the same method for the square-root, we find that 
the successive powers of ^2 + 1, namely, 2^2 + 3, 5^2 + 7, 
12^2 + 17, and so on, have their coefficients approximating to the 
1 2 5 12 
ratio of 1:^/2, thus ^ ’ 3’ y 5 jy’ etc. And if we proceed in 
the opposite direction we find similarly that the coefficients of the 
powers of ^/3 + *J/4 + ^2 + 1 approximate to the ratio of 1, 
J 2, ^4, ^/8. The computation of these coefficients may he made 
neatly as in the adjoining scheme. 
1 
4 
22 
116 
613 
3240 
17124 
90504 
etc. 
1 
5 
26 
138 
729 
3853 
20364 
107628 
etc. 
1 
6 
31 
164 
867 
4582 
24217 
127992 
etc. 
1 
7 
37 
195 
1031 
5449 
28799 
152209 
etc. 
Here the number at the head of one of the numbers in the pre- 
ceding column, thus 613 = 116 + 138 + 164 + 195. To this 613 we 
add the preceding 116 to get 729; to 729 we add 138 to get 867, 
and so on throughout. From four contiguous terms in any one 
line we may deduce the succeeding term by using the multipliers 
1, 4, 6, 4; thus 1.22 + 4.116 + 6.613 + 4.3240 = 17124. 
The numbers shown in the last column give, between 90504 and 
its double 181008, three mean proportionals 107628, 127992, and 
152209 ; and it is apparent that a fifth line deduced from the fourth 
one would be double of the first. 
