io8 Report S.A.A. Advancement of Science. 



modulus of elliptic functious connected with the length- of an arc of 

 the ellipse and the length hence of an arc of the geodesic. 



. cos Uq sin a (i—c-) cix 



. ' . urn- — 



y I -.- cos2 //o sin2 « (I _^^ gjj^2 ^^.) ^i-A2sin2.r 



To put this into Jacobi's form for the 3rd Elliptic Ijitegral, in 

 which the factor in the denominator is i — k- sn- A sn- ii\ take 



sn A =— which lies between i and-, that is take .4 = K + //}. 



Also take (,' -=w, i.c.,x=uw {ic, k), and remembering 



, -= IV + j-5 — . IT {iv, A) 



' « (I + // sin^.r) V I -yt2 sin-' .v ^^^ ^ ^n A 



where // = —k- sn- A, we get 



cos II sin a (i— f-) 



^ — 00= — X 



V I — 1'2 cos- //o sin- o 



L en (K + //3) dn (K + 1/3) * * J 



but sn (K + //3)=- . • . en (K + 7/3) = ' ^ ^ ''' , dn (K + 2/3)= ^A"Lj^'' ; 

 I' c c 



substituting these and putting in the value of k, we get 



y/i—c- cos^ II sin^ a 



5. From the relation between f and // we must have 



— -COS-// — r positive 

 III- 



i.e., a- cos- II — a- cos- //^ sin- a positive 



. • . /'- — sin- II positive 



and greatest values of // are +sin~^/>, but sin 77=/' sin r 



. • . greatest values of r are + ^, 

 '~ — 2 



. • . as we go round geodesic, the corresponding point on ellipse 

 moves completely round it. 



The whole length of ellipse then is equal to the length of the 

 geodesic from a value of 77 back to the same value. 



What is the difference between the original value of ^ and its 

 value on return to the same value of 11 ? 



