INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 



247 



considered a function of w, denoted already by = am w, instead of looking 

 at w, in Legendre's manner, as a function, F$, of <. Jacobi adopted the idea 

 in his Fundamenta nova, and employs the elliptic functions 



sin <p = sin am u, cos <j> = cos am w, A$ = A am u, 



single-valued, uniform, periodic functions of the argument u, with (quarter) 

 period K, as (/> grows from o to JTT. Gudermann abbreviated this notation to 

 the one employed usually today. 



11.2. The E. I. I is encountered in its simplest form, not as the elliptic arc, 

 but in the expression of the time in the pendulum motion of finite oscillation, 

 unrestricted to the small invisible motion of elementary treatment. 



The compound pendulum, as of a clock, is replaced by its two equivalent 

 particles, one at in the centre of suspension, and the other at the centre of 

 oscillation, P; the particles are adjusted so as to have the same total weight as 

 the pendulum, the same centre of gravity at G, and the same moment of inertia 

 about G or 0; the two particles, if rigidly connected, are then the kinetic equiva- 

 lent of the compound pendulum and move in the same way in the same field of 

 force (Maxwell, Matter and Motion, CXXI). 



Putting OP = I, called the simple equivalent pendulum length, and P starting 

 from rest at B, in Figure i, the parti- 

 cle P will move in the circular arc 

 BA J3'as if sliding down a smooth curve ; 

 and P will acquire the same velocity 

 as if it fell vertically KP = ND; this 

 is all the dynamical theory required. 



(velocity of P) 2 = 2g-KP, 

 (velocity of A0 2 = 2g- ND-sinMOP 

 : ^ -ND-NA-NE, 



and with AD = h, AN = y, 

 = h - y, AE = 2l, NE= 2l- 



FIG. i 



where Y is a cubic in y. Then / is given 

 by an elliptic integral of the form 



/ = This integral is normalised to Legendre's standard form of his 

 v Y 



E. I. I by putting y = h sin 2 <, making AOQ = <j>, h-y = h cos 2 <j>, 

 2l - y = 2l (i - /c 2 sin 2 </>), 



2l AE 



K- is called the modulus, AEB the modular angle which Legendre denoted 

 by 6; V(i K 2 sin 2 </>) he denoted by 



