or a, x, y, z, b are in geometric progression.

Let each of these ratios be equal to -. Then, multi

plying them together, 6 m, or n = (*)

and the corresponding value u = atit, is a maximum.

Multiply these equations by x, y respectively and add, then, by the original equations 1+ - = 0.

Substituting this value of 1 in the preceding equations, and grouping together the terms multiplied by the same variable,

The upper sign giving a maximum and the lower a minimum. (8) Let u = cos x cos y cos z,

with the condition x + y + z = 7. By taking the logarithmic differential, and using an indeterminate multiplier, we easily find

x = y = x = , and u = }, a maximum. (9) Find the maximum value of

u = al + bm + on, 1, m, n being variable and subject to the condition

1+ ma + n° = 1. We easily find by the use of an indeterminate multiplier that

(a + b + c)!? (a + b3 + c)!? (a? + b3 + c^)? and therefore u = (a + b3 + c?).

This is the solution of the problem, “ To find the position of the plane on which the sum of the projections of any number of planes is a maximum :" 1, m, n are here the cosines of the angles which the plane of projection makes with the co-ordinate planes.' (10) Find the maximum value of

u = (x + 1) (y + 1) (x + 1), x, y, z being subject to the condition,

a+boc* = A. Taking the logarithmic differential of both equations we have - dx dy dz

+

+ - = 0, X + 1 y + 1" x + 1

dæ log a + dy log b + dz log c = 0. Whence, by using an indeterminate multiplier 1,

and

From these we find

log (Aabc) log (4abc)
?, y+1=-
3 log a
3 log b ?

3 log c
{log (Aabc)}}
u =

log a'. log bo. log (3. This is the solution of the problem : “If a, b, c be the prime factors of a number A, to find how many times each factor must enter into it, that it may have the greatest number of divisors." Waring, Medit. Algeb. p. 344.

(1) To find the rectangular parallelopiped which shall contain a given volume under the least surface.

If x, y, z be the edges of the parallelopiped, and if a3 be the volume of a cube to which it is equal, then by the condition of the minimum we easily find

X = y = 8 = a, so that the surface equals 6a, a minimum.

(12) To inscribe the greatest rectangular parallelopiped in a given ellipsoid.

Let the equation to the ellipsoid be

and let x, y, z be the half edges of the parallelopiped, then

u = 8xyz is to be a maximum, x, y, ~ being subject to the condition

By the method of indeterminate multipliers, we easily find

a b c 8abc