224 
ME. C. ^Y, 3IEEEIFIELD OX A XEW :kIETHOD OF APPE0XI:MATI0X 
for its use is cumbrous ; but there is no other way of treating them which is not open 
to the same complaint, for, in reality, they are exceedingly complex of themselves. 
The following are the first three terms of the arithmetic means, with the correspond- 
ing harmonic means written under them, derived from x and y : — 
-1- Qxy + y^ 
2 ’ 
2xy 
A[x-\-y) 
4:xy{x + tj) 
x^ + ^xy + 7/^’ 
x"^ + 2'^x^y + 7 + 2 
8 {x^ + + ’Ixy'^ -f 
Sxy{x^ + Ix'^y +‘lxy’^ + f) 
x-\-y' x'''‘ + Gxy-\-y^' x‘^ + 2Sx^y + *i0x-y’^ + 2Sxf + i/^ 
It is convenient to use the arithmetic in preference to the hannonic series for inte- 
gration, as the divisor contains one binomial factor less, 
may be resolved into the following form : — 
The third arithmetic mean 
x-\-y 
xy 
1 
2 x + y 
1 (2+ \/2)xy 1 (2— ^ 2')xy 
Note that 
and that 
x+{^-\-2V2)y 2 x-\-{B—2V2)y 
(2+y2r=2(3+2v'2)=3::f^= 
log(3-l-2^/2) = 0-76555 13706 75726. 
If a further approximation be thought necessary, it is possible to resolve into partial 
fractions the fourth arithmetic mean ; but if we go beyond this we shall have to solve a 
reciprocal equation of the eighth degree with all its roots real. The foiuth arithmetic 
mean will have for its divisor 
1 6 H-y )( oif -\-&xy-\-y’^){x^-\-2 Sx^y - 1 - 7 Oary^ -J- 2 Sxy^ + y^) : 
the roots of the biquadratic factor, with their signs changed, are 
7 + 4^2+ ^(80+56x/2) and 7+4x/2- ri(80±56.s/2): 
their approximate values are 
25-27414 
23690 
882 
2-23982 
88088 
434 
0-44646 
26921 
718 
0-03956 
61298 
966 
The difference between any (say the wth) pair of means of the series has always [x—y) 
to the power of 2” for its numerator, and the product of the denominators for its deno- 
minator. The logarithm of the error is therefore always much less than 
2" log (x-y)—{2’^-l) log (a’+y)— log 2. 
I shall now indicate the mode of applying this method to the general form of an 
n- • r- i- ri+Nsin^<a dp , 
elliptic function, \ , , ^ . = = \ud(p. 
1 ’J l+nsimip \/l — c^sin^ip ^ 
It is obvious that the nearer the ratio a’ : ^ is to unity, the less number of terms shall 
we require to obtain a given degree of accuracy. In elliptic fimctions which inv olve a 
