linear Valuation of Surd Forms. 205 



variables ; for whilst his construction is begun in space, his result 

 subsides to a representation in piano. But between these two 

 cases there is a very marked distinction ; which is, that whilst 

 for a surd radical with two variables every change in the limits 

 proposed gives rise to a change in the corresponding linear form, 

 such is never the case with a surd form with three or more 

 variables, unless the limits be expressed by a single linear in- 

 equality between the variables which enter into the surd form, 

 and the surd form itself. Thus, for instance, if \/ a: 2 -\- y* + z* 

 is to be represented linearly within the limits z > x, z>y (for 

 greater conciseness I throughout suppose the variables to be 

 positive), the linear representation will be precisely the same as 

 for the_single limit z > Va? + «/ 2 , or, which is the same thing, 

 z — */\ Vx* + y 2, + z 2 > ; and accordingly for the problem with 

 three variables there is usually a preliminary question to be 

 solved, viz. to find the single inequality of the kind proposed 

 which involves the satisfaction of the given limits, and is 

 capable of being substituted for them without increasing the 

 maximum proportional error. This preliminary question may 

 be reduced, as will be seen, to an elementary geometrical 

 form, and is strictly tantamount to the problem following : — 

 Imagine a pincushion with a number of pins stuck into it, 

 to find the least ring which can be made to take them all in, — 

 a problem proposed by myself some four or five years ago with 

 reference to points in a plane, in the Quarterly Mathematical 

 Journal, and of which Professor Peirce of Cambridge University, 

 U.S., has favoured me with a complete solution, which is equally 

 applicable to the sphere, the case with which we shall be prin- 

 cipally concerned in what follows. 



I shall begin, then, with supposing R to be an integer homo- 

 geneous quadratic function of x, y, z, where Xj_ y, z, U are sub- 

 ject to the linear inequality Ax -f By + Cz— v'R > 0. The geo- 

 metrical solution, as such, will be seen to be equally applicable 

 to the case of two, and the analytical representation to which it 

 leads to any number of variables. 



The problem to be solved is to find a linear form Lx + My + Nz 



such that the greatest value of ^ -^ - — 1 shall have 



the least possible arithmetical magnitude, without regard to sign 

 as positive or negative, for all values of x, y, z satisfying the 

 proposed inequality. 



It is clear that, as the entire question is one of ratios, we 

 may subject x, y, z to the condition expressed by E = l without 

 affecting the result ; in other words, we may consider x, y, z as 

 the coordinates of a point limited to lie on the segment of the 



