linear Valuation of Surd Forms. 207 



the dominant is a maximum, (r + 1) out of the whole number of 

 the functions must all alike represent that dominant. Thus 

 leaving for a moment in our original problem the case of three 

 variables, and going down to that of only two variables, in which 

 case we have to deal with a curve of the second order in lieu of 

 a surface, and are to suppose that a segment of such curve is cut 

 off by a right line A, and are required to draw another right 

 line B such that the maximum square of the quotient of a per- 

 pendicular upon B from any point in the segment by the per- 

 pendicular from the centre upon B is to be a minimum, we evi- 

 dently have to solve the same problem as if we had to find 

 the least value of the dominant of three quantities involving two 

 parameters, two being the number of constants required to fix 

 the line B ; those three quantities being the squares of the frac- 

 tions whose numerators are the three perpendiculars from the 

 extremities of A, and from the vertex of the arc cut off by B 

 upon B, and. their denominators the perpendicular upon B from 

 the origin ; accordingly the line B must be so chosen as to make 

 the three perpendiculars in the numerators, without reference 

 to sign, all equal, so that B is parallel to A, and bisects the 

 sagitta of the segment cut off by A, i. e. the longest perpendi- 

 cular from any point in the segment upon A. 



In the case of R being, as originally supposed, a function of 

 x, y, z, we may take an indefinite number of points in the sec- 

 tion of the surface R = 1 made by the plane Ax + By + Cz— 1 = 0, 

 and the summit of the segment made by the plane to be deter- 

 mined ~Lx + My + N,sr=l, and may show by the same reason- 

 ing as above (there being now three parameters) that four of 

 these perpendiculars must be equal inter se 3 which proves, to 

 begin with, that at all events the two planes must be parallel ; 

 and then the reasoning applied to two functions of one parameter 

 will further show that this plane must bisect the sagitta of the 

 segment cut off by the given plane Ax + By + Cz — 1 = *. And 



* The absolute liberty of the plane sought for (Lx+My-\-T!sz=.l) to 

 take up all positions in space, and the absence of singular points in the 

 segment cut off by the plane Ax-\-By+Cz=l, suffice to show that the con- 

 ditions of variation necessary for the legitimate application of the theorem 

 employed above are satisfied. If the minimum dominant is not at one of 

 the points of equality given by the theorem, it must lie either at some mi- 

 nimum, or at all events at some singular point of one of the functions of 

 the system to which the dominant belongs, or else at some point corre- 

 sponding to the contour, so to say, if there be one, of the space within 

 which the parameters are contained. In the case before us, the parame- 

 ters, however chosen, to fix the position of the plane are perfectly inde- 

 pendent, so that there is no limiting contour ; and it is obvious that the 

 functions representing the distances concerned from this variable plane have 

 no maxima or minima values. I do not (nor ought I to) pretend to have 

 presented the theoretical principles involved in the limitation of the general 



