I said k - 1 ==== \ k + 1 k - j > B ( ) n U (n) / j + 1 j + 1 j ==== ==== k + 1 \ k j = 1 phi(n ) > t = ----------------------------------- + -----------, / k + 1 k + 1 ==== t < n (t,n)=1
where B are Bernoulli numbers, j a b c a-1 b-1 c-1 phi(p q r ...) := p (1 - p) q (1 - q) r (1 - r) ..., and a b c k k k U (p q r ...) := (1 - p ) (1 - q ) (1 - r ) ..., k
where p, q, r, ... are distinct primes. In fact, a b c a-1 b-1 c-1 phi(p q r ...) := p (p - 1) q (q - 1) r (r - 1) ...,
as in the earlier message (and which you all know). --------- I added "volume" and "surface" of segment of n-sphere to Macsyma's GeoFuncs package, e.g., (c355) block([radexpand:false],sphere_segment_curved_surface(n,r,t)) n - 1 ----- n - 1 3 n n 1 t 2 hyper_2f1(-----, - - -, - + -, ---) (2 %pi r t) 2 2 2 2 2 2 r (d355) ---------------------------------------------------- n - 1 (-----)! 2 (t is thickness. radexpand:false just prevents the exponent in the numerator from distributing.) You can also give two abscissae instead of the thickness, getting powerseries in x1 and x2 instead of t=r-x1 and r-x2. (Perhaps I should have left it an integral, but half the time, the series terminate in just n/2 terms.) Try the n=2 case: On a circle of radius r, the chord of a segment of thickness t subtends what angle? Expecting something like 2 cos((r-t)/r), or even 2 atan(sqrt(r^2-(r-t)^2)/(r-t)), I was surprised by (c356) (assume(r>0), hypersimp(subst(2,n,%))/r) sqrt(t) (d356) 4 asin(---------------) sqrt(2) sqrt(r) (c357) rootscontract(%) t (d357) 4 asin(sqrt(---)) 2 r This says that the chord from an endpoint of the original chord to the midpoint of the arc is twice the mean proportional of the thickness and half the radius. Can this be seen directly, as opposed to via two Pythagorisms? -------- Apropos RKG's pun typo, mathematicians can indeed be vectors. I infected a 7th grader with the following diseased factoid, probably caught from a confused highschool teacher, and never examined critically: :-( :-( :-( :-( :-( :-( :-( :-( :-( :-( :-( "The gravitational effect of an extended body on a point outside its convex hull is the same as if all its mass were concentrated at its center of gravity." )-: )-: )-: )-: )-: )-: )-: )-: )-: )-: )-: )-: )-: )-: The absurdity of this is evident from comparing the force at the origin due to aligned unit masses at distances 1 and 3, vs a two-unit mass at 2. Otherwise, masscons would not perturb lunar orbits, and the oblateness of the Sun would not contribute to the precession of Mercury. --rwg