] 70 
MINUTES OF PROCEEDINGS OF 
6. Examples on Interpolation. 
=«. + »a«. +a 3 *, + w •» - 1 • • - 2 
1 . 2 
1.2.3 
A%* + &c. (A) 
Find log 142; given log 15, 16, 17, &c. 
Let u K = log 140 = 2-1461280 
u x+ i = log 150 = 2-1760913 
u u + n — log 160 = 2-2041200 
u xM3Z = log 170 = 2-2304489 
u v+4l = log 180 = 2-2552725 
u x+5l = log 190 = 2*2787536 
u x+6l =. log 200 = 2*3010300 
+ A 
299633 
280287 
263289 
248236 
234811 
222764 
-A 2 
19346 
16998 
15053 
13425 
12047 
+ A* —A 4 + A 5 —A6 
2348 
1945 
1628 
1378 
403 
317 
250 
86 
67 
19 
Here, x = 140, 10. The columns of differences being all dimin¬ 
ishing are alternately positive and negative, as explained above. 
To find log 142, we must put n — \ m (A). 
App q iII >- \ log 142 = log 140 + £ X '0299633 — A 3 + ! --■A 3 — Sec. 
= log 140 + |A+*A 3 + T fyAS + ^A‘+ T fffyA8+^^.A«+ Sec. 
log 140 = =2-1461280 
JA = = -00599266 
2 ? 5 A 2 = x -0019346 = -00015477 
T f 7 A 3 = x -0002348 = -00001128 
^y y A* = T jj4 T X -0000403 = -00000135 
rfffeA 5 = = -00000022 
.-. log 142 = 2-1522883 
In like manner, putting n — f, we obtain 
See eqn (c 2 ) log 144 = log 140 + f A + ^A 2 + -jf^-A 8 -+- ^A 4 + &c. 
App, All, 
So also, putting n = J- and f successively, we obtain log 146 and 
log 148. 
Thus we have a complete interpolation. 
Log 141 may either be found by putting n = X V, or by making use of 
App m the resu ^ s already obtained, taking fresh differences and putting n = 
Before studying the practical use made of interpolation by Mr. Bash- 
forth, we must consider the main principles of his chronograph. 
Art. 56. 
