APPLICABLE TO ELLIPTIC AND ULTEA-ELLIPTIC FUNCTIONS. 
227 
If we equate n to N, we obtain for the approximate value of 1 (1 — sin^ (p)~id<p, or 'P{cp), 
Jo 
I ( 1 — ^?'r^tan-‘{(l— tan-’| ^ 1 — tan (pj 
'tan-*|(l-?::y^c^ytan?)| 
I now apply the numerical values <p= c= sin 45°, which, as has already appeared, 
lead to the most unfavourable case for approximation. For <p = \'7r all the inverse 
tangents become =p‘T, and, by reducing the last two terms to a single radical, we 
easily obtain 
F^sin 45°, |:r) =ix{|+l sin 45° + 5 sin 60° + 
1*18034 09494 53=1-85407 52150. 
This exceeds the exact value given by Legendee, 1-85407 46773, by 0-00000 05377. 
The logarithm of the factor of is 0'07200 74743, of which the error is 0-00000 01290, 
in excess. 
It follows from this that the necessary error of the process, where the third mean of 
the arithmetic series is used, will never be more than a unit in the seventh place, and it 
is evident that this error ^^'ill always be in excess. By an arbitrary correction, of which 
the amount may be easily guessed in any given case, the seventh place may always be 
made accurate. 
2 H 2 
