636 Mr. C. E. Van Orstrand : Inverse 
to be used in preference to the first when a great number of 
values is to be interpolated. In the many instances in which 
only one or two values are to be obtained, the first method is 
the most convenient. Functions determined by any of the 
preceding methods may not of course be very reliable for 
extrapolation. The last formula, for example, is applicable 
between the limits 30° and 90°, but is uncertain for values of 
*<30°and >90°. 
It is desirable to ascertain if some other arrangement of 
the terms of the reverted series would facilitate computation. 
The numerous possible combinations of the quantities are 
best exhibited by again making use of Professor McMahon's 
expression for the general term of a reverted series. Putting 
0i= -7; 62= -7; 63=-; o 4 =— ;...., 
a a a 1 a 
the (m— l)thterm of the reverted series may be written 
m(m + l)(m + 2)...(p + ^ + ../m-2))[2^^r]^- 1 ) 
in which the exponents and subscripts are subject to the 
condition pi + gj + ... = m -2. 
Giving ra successive values, say from 3 to 8, we easily find 
the terms containing all possible combinations of 6? b\ . . . b% . 
They are 
m=3 6 l9 
4 V b 2 
5 V h b 2 
h 
6 v \n 2 
h h 
b 2 b % b± 
7 bf \H 2 
h s h 
h b 2 b± b± b 5 
8 b, 6 b^b 2 
b ± n 3 
b±b 2 b^b^ b^ 5 b Y b 2 b 3 
2 b 2 * A- 
Each tennis of weight ?n — 2. There are two combinations 
of the terms of nearly complete symmetry in which functions or 
the same quantities appear in a form suitable for computation. 
Thus, assuming that the given series terminates with the fifth 
power, one arrangement of the terms is 
+ & 2 [l + 6, + V + &i 3 + ....] 
+& 3 [i+&i+V+& 1 3 +....] 
+M1 + &1 + VH-.... ] 
+& 5 [i +&!+.... ]+.... s^ +■&*+... j 
+ 
> w 
