APPLICABLE TO ELLIPTIC AND IJLTEA-ELLIPTIC EUNCTIONS. 
429 
to the 10 th figure, therefore, considered as an integer, the index is right as it stands, 
and we need not bestow thought on the proper placing of the correction, as we must if 
we use any other number of figures, 
29. If we had been content with five decimals, the calculation would have been very 
easy, for in that case we might have used six-figure logarithms, and have made ^ = 4, 
thus omitting the terms containing 22° \ and 67°^. We should get 
7-10091 3039 
1-86022 6468 
4)5-24068 6571 
1-31017 1643 value required. 
1-13483 2441 
0- 72539 4027 
1- 86022 6468 
30. It is worth while to notice a case which will sometimes occur, namely (using the 
notation of the last example), that the values may be so selected as to give, for one of 
the values of k, sin a = sin cosy, and thus each of the terms into which ^Zj,d(p was 
divided would become infinite. Of course the difficulty is only apparent ; for in this 
case the proper value is d<p — which the integral may be at once 
sin^.cosip 
found by differentiating the expression 
•' o 1 1 — sin-'a, 
sin^ip 
Section III . — Extension of the Method. 
In respect of rapid approximation and precision of limit, the foregoing processes 
leave nothing to be desired, as far as concerns the radical of the square root ; but they 
do not go beyond that. Mr. Sylvestee has given an elegant extension of the method 
to radicals of a higher index, by means of symmetric functions 
The more general problem before us is that of approximating to the integrals of 
irrational functions by means of rational substitutions. 
Let (p and be functional symbols, and 3 / a function of z; then, that (p[z).y^ and 
<p{z) should both be approximations to 9 ( 2 ), depends upon approaching unity as ni 
increases. Assuming that y^ and y, are connected by the equation y^=-<p{m, 3 /,), our 
problem is to choose so that, in the tu’st place, the approximation shall be exceedingly 
rapid, and, in the next place, that <p{z).y^ and (p{z)\y„, shall both (or at least one of 
them) be thoroughly manageable, and easily integrable. In the case of the approximants 
already given, the equation y^='^{m^yP) has been made 
I am acquainted with three general methods which effect the object more or less. 
The first is the obvious one afforded by the Newtonian approximation to the roots of an 
equation ; viz., let « be a first approximate solution, obtained by trial, of the equation 
/j:= 0 , and call f'x the differential coefficient of fx', then a second approximation is 
* See the Philosophical Magazine for December 1860, Supplementary Number, vol. xx. p. 525, note A. 
