72 UNIVERSITY OF VIRGINIA PUBLICATIONS 



■ hm^)'- 



cyclic interchanges give the other two. The Hessian of s I have not been 

 able to express in sufficiently simple form to make it worth while writing 

 down. However, we may proceed more simply by expanding s by Taylor's 

 formula. Thus let s be the value at P and s' the value at P' a point distant 

 p from P. The expansion is found to be 



s' = .s + p [acos(prJ+ p cos{Pr.^-\- y cos(pr3)+ 8 cos(p)-J] 

 + lp- — sin-(prJ + 4-sin=(/Dr3) + -^sin=(p;-J — sin-'(pr,)+| + B, 



|_ ' 1 '2 '3 ' i J 



(68) 

 where R contains higher powers of p. This shows s will be a maximum, 

 or minimum at P when the sum of the projections of a, /?, y, S (along 

 the r's) on an arbitrary liae vanishes, and the coefficient of p" retains its 

 sign for all positions of P' in the neighborhood of P. Obviously the con- 

 ditions for a minimum are satisfied when Q is inside ABCD. 



23. The segments a, p, y, 8 in absolute values must construct a 

 quadrilateral and therefore no one can be greater than the sum of the 

 other three, for an ordinary maximum or minimum. Disregarding the 

 plane at infinity the four planes 



± X ± y ± z± w = 



passing through the mid-points of the edges of ABCD contain all points 

 such that the sum of three coordinates of Q is equal to the fourth. At 

 all points inside the tetrahedron, whose plane faces pass respectively 

 through the mid-points of the concurrent edges a, i, c, f, g, h, the point Q 

 satisfies these conditions, also when Q is in the opposite diedral angles of 

 tins tetrahedron. Not all points in tlais region, however, satisfy the con- 

 ditions for an ordinarj' maximum or minimum. 



24. We seek the condition that the critical point P may occur at a 

 vertex, say D. The partial derivatives there are indeterminate, the inde- 

 termination arising through the indeterminateness of the direction cosines 



