208 Mr. C. W. Merrifield on the Geometry of 



Now the modulus ^^^ is the modulus next followincr 

 1 + sin ^ "^ 



sin 6 in Lagrange's scale (ascending), and the equation of the 



amplitudes, tan 6 = ^-t — —. — r^,, is only another form of 



^ ^ cos 2i/r + sm 6 •' 



sin (2A|r— 0) = sin 6 sin ^, 



the well-knowTi equation of Lagrange, which may also be put in 

 the form 



tan(d>—ylr)= ^ ■. — ;, tan ilr = cos ^' tan -^/r. 



^^ ^'' 1 + sm^ T T 



The reader will not fail to notice the important relation 

 ■\/r=i(^ + T), which follows from the equation, as well as from 

 the geometry of the figure. 



In the annexed figure, let the radius C A of the outer circle 

 MqM, Mg be unity, the radius Da of the inner circle r, and the 

 distance between the centres CD = e. 



Also let A Mq and Mj Mg be tangents to the inner circle, and 

 let the arcs 



Blo=2^o. AM, =2^1, AM2=2f2. 

 Jacobi has shown that these three arcs will fulfil the elliptic 

 equation 



cos ■^Q= cos yJTi cos ■\/r2+ sin ^{r^ sin -^^/^{O'^Itq), 

 provided 



1— A-Jta ,, V , 2C0S-Jrf, 



where 



Ax|r„ = (l-sin^^'sin2-.|rJ*. 



