THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 243 



The substitution (iv, } gives (z, -}, and 



1 w w I * \ 1 z z ) 



\ \ - \ ) \"/> 



12P61' = - (1 + z - z 2 ) 2 + 8z (1 - - z) 2 (9), 



s (3v) - 2(1 z)'\ is' (3r) = z (1 - z) 2 (10), 



,9 (6v) = 0, is' (6r) = z 3 (1 z) 3 (11), 



x (9r) = s 3 (1 z}, M (9r) = s 13 (1 =) (12), 



(13). 



z w v (z t) s 



18. For ^ = 23 the class of the modular equation is 2, so that simple relations 

 cannot he anticipated; but 'as the class is zero for fj. 25, it is possible that the 

 ellipsotomic equation y.,- =; may he susceptible of reduction (L.M.S., 25, p. 275). 



19. As mechanical applications of the preceding integrals of the First Stage we 

 may cite the case of LEVY'S " Elastica," mentioned in 5 and discussed in the ' Math. 

 Ann.,' 52, the Spherical Catenary (L.M.S., 27), and the Velarium surfaces considered 

 in F. KOTTER'S ' Inaugural- Dissertation ' (Halle, 1883). 



Take an umbrella with straight ribs, and hold its axis vertical, as an illustration of 

 a velarium. 



If the gore laid out flat forms a sector of a circle, then it is obvious that for any 

 other angle between the radii formed by the ribs, the edge and its concentric lines 

 form spherical catenary curves, as shown by KOTTKR'S equations. 



But with triangular gores (Kih'TKK, ' Diss.,' p. 38) the projection of the edge on a 

 horizontal plane is given by 



with t = ( ) ; and this is reducible immediately to our standard form (l), 5, by 

 putting 



= 16M 6 T = 4M-* (Wt - 2M-B)- + 16MW (M*t - M-) 

 = 4,- (s - 2M-B) 3 + 16M 1 A- ( - M 2 ) (3), 



and equating coefficients, 



2 I 2 



