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Another integral is according to § 4: 
y ««x f + + y + /«i«.**» f <JP + /i («i + «*) ^«i«. cosy—const. 
§ 25. The results become very intricate for the general case. This 
is evidently a consequence of the circumstance, that in the function R 
appear cosy and sin* y or in other words cosy and cos2y. The 
problem is considerably simplified if we suppose f x = 0, thus e 2 = 0, 
which means, that we suppose the planes XZ and YZ to be planes 
of symmetry for the surface. 
Let us again introduce £, so that 
a, = R*h^ « f = *,■*■( 1-0, 
then the last integral may be written in the form: 
) mv = p p + 1 z+T, 
so that we can perform again all integrations in finite form, and x 
and y may then be found as functions of the time. 
Osculating curved. 
§ 24. We return to the general case and shall proceed to investigate 
what becomes of the osculatiug curves. They are ellipses whose 
equations are found by eliminating t between 
i/« 
x — - - cos (nt 2 nfij) 
By changing the origin of time we see that for a definite osculating 
curve we can also find the equation by elimination of t between 
V**i 
so y represents the difference in phase. 
When y has an arbitrary value, the ellipse has an arbitrary shape 
and position. 
If y = 0 or x a straight line is described passing through 0. 
If ?P = y the axes of the ellipse lie along the axes of coordinates. 
