634 Mr. C. E. Van Orstrand : Inverse 
We disregard the algebraic sign of the last term, for a mere 
inspection of the first differences suffices to show that the 
correction to n x is positive. Following is the compulation : — 
F n = 
8-910 7867 247 
Fo = 
8-910 6617 247 
AF= 
+1250 000 
6096 9100 
a'—^+b' = 
+2577 825 
6-411 2534 
n l =AF-^-a'= 
0-484 9049 
9-685 6566_io 
r =» 1 -r a' = 
3-2744-io 
V= b J> 
2 
-768 
2-8854 
Check. 
52,r6' = 
+0-000 0701 
5-3455 
a'n =1250181 
n — 
0-484 9750 
9-685 7193 
b'n°-= -181 
0= 4° 40' 
14"-84975 
AF= 1250 000 
The value of the term 2n 1 (V6 / ) 2 is 2 x 10~ 8 , a negligible 
quantity. The formula here suggested will frequently be 
applicable when the differences are small numbers. The 
computation may be made by actual multiplication and 
division ; but in general, logarithmic computation requires 
less labour. In either case, the method just explained is 
about as easy of application as any other, except that in mis- 
cellaneous computations, some care must be exercised to see- 
that the term 2?? 1 (rZ>') 2 may be omitted. On the other hand,, 
if the higher differences need be taken into account, the com- 
plete formula has the important advantage of putting into 
evidence the terms necessary to obtain a required degree of 
accuracy. Thus, although third differences occur in that 
portion of Vega's ten-place table of the logarithmic sines for 
arguments = or > 2°, the terms in rd are negligible and 
the maximum value of 2n 1 (r6') 2 , which occurs in the imme- 
diate vicinity of 2°, is five units in the seventh decimal place. 
Consequently, the correction to the simple formula 
n = iii — n^rV 
is so small that it may be omitted except when extreme 
accuracy is required for arguments a little greater than 2°. 
A problem of frequent occurrence in many branches of 
applied science is the reversion of an empirical formula 
expressed in the form of a power series. Since the relation 
is given in the form (i), it is only necessary to substitute in 
(3) to obtain the value of the argument or independent 
variable. As an illustration, find the temperature of quick- 
silver when its volume is 1'01825409. The empirical formula 
given by Chappuis * expressing the increase o£ volume of 
* LandolUBcrnstein Tabellen, p. 209. 
