in the method of differences. 243 
tang. 2 u -f tang. 2 a;. tang. 3 w -j- }, see Delambre, Preface to 
Borda, p. 48, whence, by our expression, 
A w . tang. a7=sec.*a:| A”(ang. o-j-tang. a? . A w . tang. 2 o-|- tang.’#. 
A”. tang. 3 0 -j - 1 
Ex. 3. L. sin. (a? + u ) — L. sin. a? -f- L. cos. u -f- 
M [ Cot. x. tang, u — ^-'cot.*#. tang. 2 u-\~ — cot. 3 a? tang. 3 «-f- j 
Delambre, Preface to Borda, p. 45; comparing with (7) 
and (8 ) we find A w . L . sin . a? = A" . L . cos . 0 -f- 
M | Cot.xA w .tang.o — ^-cot 2 .a:A w .tang. 2 o-}--jCot. 3 a7. A^.tang. 3 *?. j 
In like manner, because 
L . cos . (#+«) = L . cos. .r-j-L . cos. u — M jtang. x. tang, u- {- 
~ tang . 2 x. tang. 2 « -}- y tang . 3 x. tang. 3 z/ -f- j 
A n . L . cos. x = A”. L . cos. 0 — M j tang, x A n . tang. 0 -j- 
~ tang.* x A n . tang. 9 o + ~ tang. 3 # A n tang. 3 o -f-* } 
Then, because L . tang. == L . sin. — L . cos., A". L . tang. = 
A”. L . sin. — A” . L . cos., therefore 
A*. L . tang, x = M { ( Cot. a? -f* tang, x) A”, tang. 0 — 
— (cot.® x — tang. 2 a:) A”. tang. 2 o -f- 
(cot, 3 x -J- tang. 3 x)A n . tang. 3 0 — j 
If these forms were to be used for interpolation, we should 
have to calculate, before the commencement of a Table, 
A L . cos. 0, A 2 . L . cos. 0, See. ; A . tang. 0, A 2 . tang. 0, &c. ; 
A . tang. 2 o, A 2 , tang.' 0, &c., &c. These latter quantities are 
to be multiplied by M, and will then serve for calculating the 
whole Table. 
If three differences are sufficient, we have, making u=i l * 
* It is the decimal division of the circle which is supposed here. 
