588 
PROBLEMS AND SOLUTIONS. 
[485 
[A r ol. ix., January to June, 1868, pp. 20, 21.] 
Note on Question 2471. By Professor Cayley. 
In the singularly beautiful solution which Mr Woolhouse has given of this 
question (see Reprint, vol. viii. p. 100), it is important to note what is the analytical 
problem solved, and how the solution is obtained. Considering a plane area bounded 
by any closed convex curve, and in it three points P, P', P", Mr Woolhouse investi 
gates the average area of the triangle PP'P", viz. this depends on the sextuple integral 
j ± [ x y" — x 'y' + x "y — x y" + x y' — x y] dx dy doc dy dx" dy", 
where the sign + has to be taken so that ± { } shall be positive, and where the 
integration in respect to each set of coordinates extends over the entire closed area; 
the difficulty is as to the mode of dealing with the discontinuous sign. It is remarked 
that the integral is 
= 6 f± \x'y" — x'y 4- x"y — xy" + xy' — xy) dx dy dx' dy' dx" dy" ; 
the variables in this last expression being restricted in such wise that x, x", x are in 
the order of increasing magnitude; the term + { } is of the form ± {x' — x ) (y" ~ &)> 
where /3 is independent of y, and where (as is easily seen) if v", u" be the upper 
and lower ordinate corresponding to the abscissa x", then /3 lies between the values 
u" and v". But x — x is positive, hence the sign + must be so taken that ± {y" — /3) 
shall be positive, that is, from y" = u" to y" = /3 the sign is —, and from y" = ¡3 to 
y" = v" the sign is +. 
Hence for the integration in regard of y" we have 
j ± (;y" ~ P) dy" = J^+ 0)" ~ I 3 ) dy" + J* * - (y" - (3) dy", = i (v" - /3f + i (¡3 - u'J; 
and the discontinuous sign ± is thus got rid of. The remaining integrations are then 
effected in the order x", y, y, x, x, the limits being for x" from x to x', for y from 
u' to v', and for y from u to v (if the upper and lower ordinates corresponding to 
the abscissa x and x' are v, u and v', v! respectively) and finally for x' from x to the 
maximum abscissa, and for x from the minimum to the maximum abscissa. The final 
result involves only single definite integrals between the extreme values of x, the 
functions under the integral sign containing indefinite integrations from the same 
arbitrary inferior limit, say x = 0; the form of the result (previous to its simplification 
by taking the axes to be principal axes through the centre of gravity of the area) is 
however somewhat complicated; and it would not be easy to show a posteriori, that 
the value is invariantive, that is, independent of the position of the axes: that this 
is so is of course apparent from the original form of the integral.