7. The maximum parallelogram of given perimeter and angles is equilateral.

8. If P1, P2, P3 denote the perpendiculars from any point 0 on the sides of a triangle, the maximum value of P12, is 8▲3/27abc, and O is then the centroid of the triangle. (By Ex. 5.)

[Otherwise thus:-Since 4abp, p=(ap1+bp2)2 - (ap1-bp) for any point on the base c, PiP is maximum when ap1-bp, vanishes, since ap1+bp equals 24. Then O is the middle point of the base. Now if P3 be supposed constant, O is on the median through C. Similarly by regarding p1 as constant, O would be found on the median through A; and so on. Therefore if the three perpendiculars vary, their product is a maximum for the point of intersection of the medians.]

15. Theorem.-If a right line be divided into any two parts a and b the sum of their squares is a minimum when the line is bisected.

Hence a2+b2 is minimum when a-b is minimum, because a+b is constant; that is when a=b.

COR. The sum of the squares of the segments of a line is a minimum when the segments are equal.

16. Problem.—If a right line be divided into any number of parts a, b, c ..., to find when

Let the segment a be divided into a equal parts; each part is therefore a/a and the sum of squares of the parts

2

= a(a)2 = a2

α

Similarly if the segment b be divided into ẞ equal parts the sum of squares of the subdivisions = b2/B; and so on.

Hence the above expression denotes the sum of the squares of the subdivisions of the parts a, b, c ..., and is therefore a minimum when these are equal; i.e., when

1. Divide a line into two parts a and b such that

2. To find a point P such that the sum of the squares of its distances, x, y, z, from the sides of a triangle is a minimum.

[Let A1, A2, Ag denote twice the areas of the triangles subtended by the sides of the given one at the point. Now since A1 = ax, A=by, and A,=cz,

2

..(1)

.(2)

..(3)

α b

This result may also be seen from the identity

(a2+b2+c2)(x2+ y2+z2) - (ax+by+cz)2

with which the student should be familiar.]

NOTE. This point is termed the Symmedian Point of the triangle, as it is obvious from (3) that the lines joining it to the vertices of the given triangle make the same angles with the sides as the corresponding medians; also since

3. Find a point P such that the sum of squares of its distances from the vertices of a triangle may be a minimum.

[If CP be supposed constant while AP and BP vary, the point P describes a circle around C as centre, and if M be the middle point of the base A P2+BP2=2A M2+ 2MP2. Hence AP+BP2+ CP2 is minimum when 2PM2+ CP2 is minimum, since AM is constant. Therefore P is a point on the median CM such that CP/PM-2, i.e., the centroid.

Similarly by supposing AP or BP to remain constant we find the same point. Hence the centroid is the required point when AP, BP and CP all vary.]

SECTION II.

METHOD OF INFINITESIMALS.

17. It has probably been observed in the preceding section that the positions of maximum and minimum of a quantity, varying according to a given law, are symmetrical with respect to the fixed parts of the figure. Thus when the base and vertical angle of a triangle are given, the altitude, rectangle under sides, area, etc., etc., are maxima when the triangle is isosceles.

In Art. 9 the triangle of maximum area is found by placing the two given sides at right angles:

Again, a figure of given perimeter and of maximum area is circular. As the variable line AB in Art. 11 rotates in a positive direction around P, according as PB recedes from the perpendicular from P on BC, the segments AP and BP approach an equality, and the triangle ABC is a minimum when AP-BP.

18. The several parts, of a geometrical figure which varies according to a definite law, can always be expressed in terms of the fixed parts of the figure and those quantities which are sufficient to define its position.

Take for example the figure of Art. 8. In any position of the vertex C, by assuming the triangle to be of given altitude; the variable parts, a, b, area, and other functions of the sides or angles can be found in terms of the base c, vertical angle C, and altitude.

Thus the variables may be regarded as functions of the given parts and the co-ordinates of their position.

It follows, then, that if the latter vary continuously those functions must do likewise.* Hence a very small change in position will cause a very slight change or increment in the magnitude of the function. Suppose in Art. 8 the circle to be divided into an indefinitely great number of equal parts, and let the vertex C occupy each point of section from A towards B. As the altitude thus receives indefinitely small increments so does the

area.

Let AB be the base of a triangle and any curve CCC,

the locus of its vertex.

* See Burnside and Panton's Theory of Equations, Art. 7.

C

In the figure as the vertex approaches C on the curve from left to right the intercept AX made by the perpendicular may be taken as the co-ordinate of its position, since if AX is known the position of C is also known.

Thus while AX continues to receive positive increments, the area, altitude, and other functions of it are sometimes decreasing, as from C to C1, and sometimes increasing, as from C1 to C2.

At the points C, C1, C2 the increments in the altitude alter in sign and therefore consecutive values are equal. Here also the tangents to the curve are parallel to the base AB, and at any other point C2 the increment of the variable divided by the corresponding increment in the function = cot a, where a is the angle made by the tangent at Cn with AB. We have seen that if AX denote the value of a variable in any position, and CX any function of AX, when the function passes through a maximum or minimum its two consecutive values are in each case equal to one another.

Suppose, for example, that a variable chord XY of a circle moves parallel to a certain direction; it gradually increases in length as it approaches the centre and if XY