653] 
ON A TORSE DEPENDING ON THE ELLIPTIC FUNCTIONS. 
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The analytical theory is very simple. Taking x, y, z as functions of the length 
s, we have 
'dx\ 2 t (dy\ 2 t (dz\ 2 
,dS; 
Js) + ils) +( ^ J ~ 1; 
the condition, which expresses that the radius of absolute curvature is = a, then is 
ff 
'd 2 x\~ /d-yV /d 2 z\ 2 _ 1 
\dsV ^ \ds 2 J + \ds 2 ) a 2 ' 
By what precedes, the point (x, y, z) may be taken to be upon a given surface, say 
upon the cylinder x 2 + y 2 = a 2 ; and we may then write x = a cos 6, y = a sin 6. Taking 
instead of s any independent variable u whatever, and using accents to denote the 
derived functions in regard to u, the equations become 
x' 2 + y' 2 + z' 2 
= s' 2 , 
x" 2 + y" 2 + z" 2 -s" 2 = - 9 s'\ 
a- 
x — a cos 6, y = ol sin 6. 
From the last two equations we obtain 
x' 2 + y' 2 = <x 2 d'\ x' 2 + y" 2 = a. 2 (0" 2 + 0' 4 ), 
and the first two equations thus become 
a 2 0' 2 + z' 2 = s' 2 , 
a 2 (0" 2 + 0'*) + z” 2 - s'' 2 = \s\ 
a 2 
and from the first of these we find 
a 2 0'0" + zz" 
s = 
whence the second equation is 
a 2 (0" 2 + 0'*)+z" 2 
or reducing, this is 
(ol 2 0' 2 + z' 2 f ’ 
(a 2 0'0" + z'z") 2 _ (a 2 0' 2 + z' 2 ) 2 
(cl 2 0' 2 + z' 2 ) ~~ a 2 
(cc 2 0' 2 + z' 2 ) (6" 2 + 0'*) + (0' 2 z" 2 - 20'0"z'z" - ol 2 0' 2 0" 2 ) = 4- (a 2 0' 2 + z'J. 
era* 
Taking here 0 as the independent variable, we have 0' = 1, 0" = 0, and the equation 
becomes 
CL OL 
or, what is the same thing, 
z "ï= (tf +z 'J-(a 2 + z' 2 )\ 
a 2 a 2 
lat 
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¡nt 
Write here 
a 2 + z' 2 — H 2 , 
10—2