linear Valuation of Sw*d Forms. 221 



The only real difficulty in extending M. Poncelet's method in 

 the manner pursued in the above unpretending study, con- 

 sisted in forming a clear preconception of the mode in which 

 any given system of limits require for the purpose in view to 

 be regarded, viz. as enveloped, so to say, in a single condition 

 (no wider than absolutely necessary) expressed by a linear equa- 

 tion between the given surd function and the variables which 

 enter into it. 



I may in conclusion just observe that if the relative values of 

 the variables be limited, not by a system of conditions giving 

 rise to a polygonal area of limitation, but by a condition ex- 

 pressed by the positivity of a single homogeneous function of 

 the variables of any degree, the variable point will then be limited 

 by the intersection of the sphere with a cone, and we should 

 have to solve a preliminary geometrical problem of circumscribing 

 a spherical curve by the least possible circle, — a question which I 

 have neither leisure nor inclination to discuss, but to which I 

 believe Mr. Cayley has paid some attention. 



Before taking final leave of my readers and the subject, I 

 devote a word to the inverse case of Three Rectangular Forces. 

 This is the case where the resultant and two of the rectangular 

 components are given, and it is the third component which is to be 

 expressed linearly in terms of them. In this case an approximate 

 expression is to be found for </z 2 —y 2 —x 2 , and the geo- 

 metrical locus which replaces the sphere becomes an equi- 

 lateral hyperboloid of revolution of two sheets. 



If the variable point be supposed to be limited to a segment of 

 one sheet of the hyperboloid cut off by the plane Ax + By + Cz = 1, 

 the discriminant of z^—y^—x 2 being 1, and its polar reciprocal 

 of the same form as itself, the approximate linear form of the 

 surd becomes 



2C* 2B?/ 2A# 



v/C 8 -B 2 -A s + l "WCF-Bs-A 8 *! ^/C 2 -B 2 -A 2 + Y 



Li • M 1 l~y/C 2 -B^A 2 



with a maximum proportional error — ^^^ ==r. 



To envelope, however, any given arbitrary system of inequali- 

 ties between the coordinates x, y, z on. the hyperboloid within a 

 single condition, Ax + By + Cz — 1 > becomes a geometrical 

 problem of somewhat greater difficulty than the corresponding 

 one for the sphere, and I do not propose to enter upon the dis- 

 cussion of it here. 



I shall content myself, as M. Poncelet has done in the corre- 



