344 
Proceedings of Royal Society of Edinburgh. [sess. 
We have to enquire how often D may be contained in C. Now the 
quotients among them are not affected by any other alteration in value, 
provided they be all changed in the same ratio; therefore, exactly as 
in the analogous well-known operation for square-roots, we seek 
some multiplier which may render D rational. This multiplier is 
evidently 4/4 + f 2 + 1, and the new proportionate values of C, D, E 
become C' = fi + 4/2 + 1, D' = 4/2, E' = 1. Here we observe that 
D' is contained in if 4 + f 2 twice, while E' is contained in 1 once, 
wherefore we write C' = 2.D' + l.E' + F' giving F' = fi - 4/2 ; and 
thereafter putting D' = l.E' + O.F' + G' we find G' = f2 - 1. Here, 
again, we see that the E', F', G' are transcripts of the previous 
C, D, E, and that thus the group of quotients 
i 2 • 1 ’ 1 1 
1 1, 0, 1 / 
must continually recur. This is concisely shown in the subjoined 
scheme. 
A= ^4 
B = 4/2 
C= 1 
D= 4/4- 4/2 
E = 4/2-1 
F 
G 
H 
I 
A = l.B + 0.C + D 
B = l.C + 0.D + E 
4/4+ 4/2 + 1 | C = 2.D + l.E + F 
4/2 (D-1.E+0.F+G 
1 4/4+ 4/2 + I | E =2.F + l.G. + H 
4/4- B 4/2 )F = 1.G + 0.H + I 
4/2 - 1 1 and so on. 
4 / 4 - 4/ 2 
4 / 2-1 
It follows from this, that the values of F and G are less than 
those of D and E in the ratio of 4/2 -1:1; while those of H and I 
are again less in the same ratio, so that the series D, F, H 
as also E, G, I form geometrical progressions having the 
common ratio f2 - 1 ; and, writing for shortness’ sake e= f2 - 1, 
we have D = e 4 / 2 , E = e ; F = e 2 4/2, G = e 2 ; H = e 3 f2, I = e 3 ; and 
so on. 
By successive substitutions we obtain the simultaneous values of 
A, B, C as shown in the subjoined scheme. In the lower part of 
it the numerical coefficients alone are written. 
