66 
[724 
724. 
ON THE DEFORMATION OF A MODEL OF A HYPERBOLOID. 
[From the Messenger of Mathematics, vol. vm. (1879), pp. 51, 52.] 
The following is a solution of Mr Greenhill’s problem set in the Senate-House 
Examination, January 14, 1878. 
Prove that, if a model of a hyperboloid of one sheet be constructed of rods 
C( 
representing the generating lines, jointed at the points of crossing; then if the model 
be deformed it will assume the form of a confocal hyperboloid, and prove that the 
trajectory of a point on the model will be orthogonal to the system of confocal 
hyperboloids.” 
Let (x 1} y 1 , z-i), (x 2 , y 2 , z 2 ) be points on the generating line of 
then 
fV 4_ K _ fl - i 
a I 2 b 2 c 2 
#i + Vl _ h. 
a? b 2 & 
1, 
I yhy-2 z \ z i | 
a 2 ¥ ¥ ~ 
or, what is the same thing, if 
x 2 y., z. 2 
a’ b’ c 
Pi, q-i, r,\ 
Pi + qi - n 2 = i, 
pi + qi - ri = l, 
PiPi + q#» - rj\ = 1. 
then