INTRODUCTION TO THE TABLES OF ELLIPTIC 



FUNCTIONS 



By SIR GEORGE GREENHILL 







In the integral calculus, / = and more generally, / 



J v X J p + Q 



where M , N t P, Q are rational algebraical functions of x, can always be expressed 

 by the elementary functions of analysis, the algebraical, circular, logarithmic or 

 hyperbolic, so long as the degree of X does not exceed the second. But when 

 X is of the third or fourth degree, new functions are required, called elliptic 

 functions, because encountered first in the attempt at the rectification of an 

 ellipse by means of an integral. 



To express an elliptic integral numerically, when required in an actual 

 question of geometry, mechanics, or physics and electricity, the integral must 

 be normalised to a standard form invented by Legendre before the Tables can 

 be employed; and these Tables of the Elliptic Functions have been calculated 

 as an extension of the usual tables of the logarithmic and circular functions of 

 trigonometry. The reduction to a standard form of any assigned elliptic integral 

 that arises is carried out in the procedure described in detail in a treatise on the 

 elliptic functions. 



11.1. Legendre's Standard Elliptic Integral of the First Kind (E. I. I) is 



d<t> 



/ 

 - 



defining <f> as the amplitude of , to the modulus AC, with the notation, 



(p = am u 



x = sin 4> = sm am u 

 abbreviated by Gudermann to, 



x = sn u 

 cos </> = en u 

 A <j> = V(i - K 2 sin 2 0) = A am u = dn u, 



and sn , en u, dn u are the three elliptic functions. Their differentiations are, 

 dd> d am u , 



d sin d> d sn u 



~ = cosd>-A<i> or = 



du du 



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