APPLICABLE TO ELLIPTIC AND ULTEA-ELLIPTIC FUNCTIONS. 
431 
The coefficients A may now be found in a variety of ways ; by solving equations (4.) 
or (5.), by simple division, or by Aebogast’s processes *. The object of the preceding 
lemma is to connect the quantities with the coefficients of division and of recurring 
series. Our results in any way are, 
ju, a^fi 
, ■ ' 
2? 
<?gjK.^ + glV 
a^jJ? + 
+ (2ff3ai + + «i'‘' 
If for Ua, «i, &c. we substitute the coefficients of the binomial theorem, so as to make 
N=(a+a?)”, we obtain 
a-\-Xi= 
(n— l)a'* + N 
na’* 
a, 
(ra-l)a" + (w+l)N 
^ (n+ l)o” + (ra— 1)N ■ ’ 
(n^ — (4n® + 2)a"N+ — 1)N® 
(71 + l)(n + 2)g^" + 4(n® — l)a”N+ (w— — 2)N^ ‘ 
Making n=2, we obtain Mr. Stlvestee’s approximants to the square root, andA„is then 
the coefficient of a?” in the development by ascending powers of 
1 
(N — a^) — 2ax—x''^ ’ 
and so far the method agrees with the Newtonian approximation by continued fractions ; 
but from this point the two methods diverge. For ?i=3, A„is the coefficient of a;” in the 
development of 
1 . 
— (^)—3a^x—Sax'^—a^ ’ 
and the successive approximants are 
2a^ + N a® + 2N 4ffl®+ 19a^N + 4N® 5ffl^ + 45o®N + 30a®N^ + N^ ^ 
3a3 • 2a3^ ' 10a6+16«3N+ N" ’ ISa^ + Slc^N + lSc^N* ' ’ 
while the second approximant obtained by successive substitution is 
1 6a9 + 5 1 fl«N + 1 2g®N^ + 2N3 
36a9 + 36a«N + 9g3JN2 ' 
What these methods all effect is simply a rational approximation to the value of y in 
du 
the equation <p(^, z) = 0. Then, making we have only to integrate in order to 
find the value of u. They thus constitute a means of approximately solving, in respect 
of w, differential equations of the form ^^=0; but they do not effect the solu- 
* See his ‘ Calcul des Derivations,’ pp. 26, 29 ; or De Moegax, ‘ Diff. Calc.’ p. 331. 
