56 DNIVEKSITY OF VIRGINIA PUBLICATIONS 



Eeturnmg to the ease ^dlen a, /?, y is ou the arc of (31) and 

 A' -\- A ^= TT, when the critical point is at .-L, we examine the function s 

 there. Let p be the small distance of a point P' from P. 



Indicating the angles which P' P = p makes with the radial coordinates 

 '■]) ''-2, ''3 of P, hj {pi\), {pr„), {pi\) respectively, we expand the value 

 of •?' at P' in terms of s at P and in powers of p by Taylor's theorem 

 and find 



s' = s-\- p[acos{p i\ )+ 13 cos{pu)-\- y cos(pn)] 



+ iP'' -^sin-(?J''-i) +^sm-(pi\) + -j?^ sin^ {p r^U + P- 



(30) 

 A necessaiy condition for maximum or minimum at an ordinary point P 

 is, as found before, 



a cos(p 'i\)-\- P cos(p i\)-\- y COS ( /; r„)= 0, 

 for all directions of p. A sufficient condition is that 



-^ sin- (p ?-i) + -^ sin= (/J /■„) + — sin"'(p''3), 

 '"i ''2 ''3 



shall preserve its sign for all directions of p. If three segments a, j8, 7 

 are in equilibrium at a point the sum of the projections of any two on 

 any line is at most equal to the tliird, as is easily demonstrated. 



In particular let P be at .1, then the value s' of s in the neighborhood 

 of A is 



s' = S-\- p[a + /3 COS{cp)-\- y C0S(Z( p)] 



+ ip"- lJ^sin^cp)+'^-sm^pb)'\ +E, (31) 



by taking, in (30), P' on A P then sm{pi\)= 0, cos(prJ=l, then 

 moving P to A. Since (c p)^{b p)= A, the gi-eatest value of 



/3 cos(cp)+ y cos(6 p) (33) 



occurs when 



/8sin(cp) — y sin(6 ;j)= 0, 



that is when p is along the diagonal of a parallelogram whose vertex is A 

 and whose sides are /8 and y laid off along A B, A C respectively. In fact 

 if V be the critical value of (32), on squaring and adding 



•f- = ;8= + y= + 2 ,8 y cos 4, (33) 



but a-' = ;3- + y- + 2 ^ y COS A, (34) 



land therefore v = a. The second derivative of v at the critical value 

 is — a and since a is positive, v is a minimum. 



