310 Mr. J.J. Sylvester : Meditation on the 



mate representation of K/a + bx + cx*. Suppose f-r-gx is the 



first approximation, as found by Poncelet's method, with a maxi- 



i . ^ (/+<7*) 2 + {a + bx + cx*) ... , 

 mum relative error e. then -¥ — * ', » ; — will be a 



much closer approximation, with a relative error never exceeding 



e* . e 2 



pt — in excess, nor « in defect. So a still nearer approxi- 



2 + e 2—e rr 



. n , (f-\-gx) 3 + S{/+gx){a-hbx-hcx 2 ) ., .-.£ 

 mation will be w * ' i , x ~ , * ': — - — « \ with a relative 



3(f+gx)* + a + bx + cx 2 



e 8 e* 

 error never exceeding ~-r — ~ — jc- Q in excess, nor - — ^ jr-s in 



° 4 + 6e + 3e 2 4— 6e-f-3e* 



defect, and so on. The marvellous facility which these formulae 

 afford for the calculation of elliptic and ultra-elliptic functions, 

 and not merely for their computation as by a method of quadra- 

 tures, but (which is of far greater importance) their quasi-repre- 

 sentation under circular and logarithmic forms, with assignable 

 limits of proportional error, will be illustrated in a future com- 

 munication. As regards the idea of substituting rational for 

 irrational functions, I have only to-day learned from Mr. Cayley 

 that I am anticipated in this by Mr. Merrifield*, in a paper very 

 recently read before the Royal Society, but not yet printed in the 

 Transactions f- 



* I quite concur with Mr. Merrifield, and in fact before being made ac- 

 quainted with the existence of his paper, had emitted the same opinion 

 (among others to Dr. Borchardt of Berlin), that the substitutive method, 

 consisting in the employment of rational functions in place of the radical, 

 affords by far the most expeditious means for the calculation of elliptic 

 functions of all orders, especially the third, and supersedes the necessity 

 for the construction of special auxiliary tables. I believe, however, that 

 my substitutions, founded on Poncelet's views, are in general the best that 

 can be employed for the purpose. In addition to other advantages they 

 possess this, which deserves notice — that as we know a priori a superior 

 limit to the proportional error, the arithmetical values of the integrals to 

 which they are applied may be brought out correct to any required place 

 of decimals, without its being necessary to calculate and compare a superior 

 and inferior limit to the integral, either one of these being sufficient 

 in my method to indicate its own reliable degree of precision. 



t In general it is obvious, if <f>x between the limits a and b retain always 

 the same sign, and tyx within these limits be sometimes greater and some- 

 times less than (f>x, but the difference between them be always less than 



f <f>x, then I dxtyx will differ from 1 dx<f>x by considerably less than 



/•a Jb Jb 



( I dx(f>x. Paradoxical, however, as it may at first sight appear, there are 



extreme cases where this difference tends to a ratio of equality with 

 < 1 dx<\>x. The complete elliptic function of the first order may be made 



