290 Proceedings of Royal Society of Edinburgh. [june 20 , 
which lie all on one side of the absolute ratio, being indeed the 
alternates of the preceding. 
Now, in forming a list of the successive derivatives of x for 2 = 1 , 
according to the law above explained, and writing for clearness’ 
sake, A for *22389, B for *57672, we get the progression 
X 
X 

+ 
A 
B. 
,x 
rr 
— 
A 
+ 
B. 
X 
= 
+ 
2 A 
— 
B. 
,x 
= 
— 
5 A 
+ 
2 B. 
,x 
= 
+ 
18 A 
— 
7 B. 
,x 
= 
— 
85 A 
+ 
33 B. 
,x 
&c., 
+ 492 A 
&c. 
191 B. 
But these coefficients of A and B are developed exactly as are 
the members of the approximating fractions, so that since A : B is 
nearly as 191 : 492, the difference 492 A-191 B must be small, and 
must continue to decrease as we proceed farther. Hence, if we can 
show that the above progression of quotients 1, 2, 3, 4, &c., neces- 
sarily holds good, we shall have demonstrated that the progression 
of derivatives never becomes divergent. 
If we treat the progressions A = x, B = — lz x by the method for 
continued fractions, taking the excesses, and dividing each excess 
by z, we get at once the following results : — 
A = 1 - 
B = 1 - 
-A+ B = zC ; 0 = ^- 
-B + 2C = zD; D = r ^-g - 
-C + 3D = £E; E = 1 g ^ 4 " 
z z 2 z 3 
T 2 + 1 X 2 2 ~ l 2 . 2 2 .3 2 + 
z ' z 2 z 3 
JA2 + l 2 . 2 2 .3 "... ii 2 .4 + 
Z Z 2 z 3 
l 2 . 2 . 3 + . . 2 2 . 3.4 . . 3 2 .4.5 + 
Z Zz 2 
H.2.3.4 + .. 2 2 .3.4.5 
z z 2 
l 2 .2.3.4.5 + ..2 2 .3.4.5.6 
- &c., 
- &c., 
and so on ; wherefore, in general, the formation of the fractions 
