614 
PROBLEMS AND SOLUTIONS. 
[705 
or, what is the same thing, the coordinates of the six given points may be taken 
to be 
+ zp x — 2w )B X V ^ 
2Z'p — onnr\- \ ^ X7 
3( xp 3 
+ ( 
+ ( xp B - 2ypi 
+ (2 xp 6 - yp 5 
[Yol. lxl, p. 123.] 
3185. (Proposed by Professor Cayley.)—An unclosed polygon of (m+1) vertices 
is constructed as follows: viz. the abscissae of the several vertices are 0, 1, 2, ..., w, 
and corresponding to the abscissa k, the ordinate is equal to the chance of m + k 
heads in 2m tosses of a coin; and m then continually increases up to any very 
large value; what information in regard to the successive polygons, and to the 
areas of any portions thereof, is afforded by the general results of the Theory of 
Probabilities ? 
[Vol. LXL, p. 124.] 
3229. (Proposed by Professor Cayley.)—It is required to find the value of the 
elliptic integral F (c, 6) when c is very nearly = 1 and 6 very nearly = \ir; that is, 
the value of 
rfrr-a- d6 
J 0 {1 — (1 -& 2 )sin 2 0|i’ 
where a, b are each of them indefinitely small. 
N.B.—Observe that, for a = 0, b small, the value is equal log 4¡b, and for 6 = 0, 
a small, the value is — log cot