1903.] Integral and the Mlipsotomic Problem. 3 



(11) Xi, X 2 = l±(o~ 1 -o)x-x 2 , 



where o denotes the octahedron-irrationality, o = ,Jk. 



But with /x = 8n f this changes to 

 (12) (D - x 2 ) n exp. 2nl (v) i 



= [E + E^ + . . . +(- l)^-i] ^JXi 



+ i [E -E^+ ... - ( - l)^-i] 

 (13)* X 1? X 2 = l±(o~ 1 + o)x + x 2 . 



The results are worked out in the memoir for numerical values 

 as far as possible, starting with the simplest, 3, 4, 5 ... , and the 

 application is shown to other associated mechanical problems, such 

 as central orbits, the spherical catenary, the elastica and velarium. 



Provided with a list of these integrals, the student of Applied 

 Mathematics will be able to effect the complete discussion of many 

 mechanical problems now abandoned in an unfinished state; at the 

 same time the exploration along the simplest line of progress is 

 effected of the general analytical field, and mathematical research is 

 guided along a road likely to lead to useful development in the theory 

 of elliptic functions. 



Incidentally the elliptic section or division values (Theilwerthe) are 

 determined, as well as those of the zeta and theta ^functions, as 

 algebraical functions of a parameter, in a form such that 



are of simple algebraical character; and it is shown that this, the 

 Ellipsotomic Problem, as it may be called by analogy, depends 

 essentially on the discussion of the curve given by (4), which may be 

 called the ellipsotomic equation and curve in the co-ordinates x and y, 

 or on that of a reduced form, involving the determination of its class, 

 and the expression of its co-ordinates as functions of a parameter. 



The coefficients in the Transformation of elliptic functions are 

 symmetric functions of these section values, so that the Transformation 

 may be considered as determined incidentally, but as the Transforma- 

 tion theory has no utility in dynamical applications, this branch of 

 pure analysis has not been pursued. 



(14) 



and 



B 2 



