210 Mr. J. J. Sylvester on Poncelet's approximate 



or 



K=j(v/A a +B* + C*+1), 



that is to say, the approximation is 

 2A 



14Va*Tb«+~c 2 * +&c -' 



the maximum error being 



y^A»+B«+O a -l 



which is easily seen to agree with the general formulae above 

 given. 



When, as is usually the case in applying these results, the 

 plane A#-fB?/-f- Cz— 1 = is not directly given, but is to be 

 found as the plane passing through three given points whose 

 coordinates are a, b, c; a', b' 3 c' ; a", b n } c" respectively, we may 

 use the equations 



where 



But it may also sometimes be needful in practice, as will pre- 

 sently appear, to determine the plane with immediate reference 

 to only two points upon the surface. 



Application to the surd form which represents the resultant of 

 three forces at right angles to each other. 



Here R = \Zx* + y 2 + -z 2 , and R = 1 represents a sphere. Two 

 cases will be shown to arise. The first, the more frequent one, 

 is that already alluded to, where a limiting plane has to be drawn 

 through three given points. For this case, using F, G, H in 

 the sense in which they have immediately above been employed, 

 the linear representation of y/x^+y^+z* becomes 



2F 2G 2H 



Q+N^ + Q + N^+Q+N*' 



