[724 
724] ON THE DEFORMATION OF A MODEL OF A HYPERBOLOID. 67 
RBOLOID. 
Similarly, if (&, y x , £i), (£ 2 , y 2 , g 2 ) be points on generating line of 
« 2 /3« X> 
and if 
a’ 1’ 7 = Pl> qi ’ ri ’ 1’ 1’’ 7 = P 2 ’ q2 ’ rs; 
then 
Pi 2 + qi 2 - L 2 = 1, 
p 2 2 + q 2 2 - r 2 2 = l, 
Pip 2 + qiq 2 - r 2 r 2 = l. 
Hence if (w x , y x , z x ), (&, y x , £,) be corresponding points on the two surfaces, that 
is, if 
x \ Vi z \ £i Vi £i 
a’ b’ c ~ a’ /3’ 7’ _Pl ’ ?1 ’ ri ’ 
and similarly, if (a? 2 , y 2 , z 2 ), (£ 2 , y 2 , Q are corresponding points, that is, if 
52.] 
5 St Vi i’ =D „ r . 
a ’ b ' c a’ /3’ y '' 2 ’ 
: Senate-House 
then we have, as before, the system of three equations 
•ucted of rods 
n if the model 
rove that the 
m of confocal 
Pi + qi 2 - n 2 = 1, 
pi + qi - r? = 1, 
P1P2 + qiq 2 - nn = 1. 
Then if the two surfaces are confocal, that is, if 
a 2 , /3-, — 7 2 = a 2 + A, 6 2 + A, — c 2 + A, 
we shall have 
(Ph ~ x -if + (?/i “ Vif + (*i ~ z if = (£ ~ I2) 2 + Oh - %) 2 + (Si - &) 2 . 
For this equation is 
a 2 (jpi - pit + & 2 (?i - qif + c 2 (n - r 2 ) 2 = a 2 (p a —y> 2 ) 2 + /3 2 - g 2 ) 2 + 7 2 (r, - r 2 ) 2 , 
that is, 
(pi ~pif + (?i - ? 2 ) 2 - (n - r 2 ) 2 = 0, 
an equation which is obviously true in virtue of the above system of three equations. 
Hence, if on confocal surfaces 
a 2 y 2 s 2 f v 2 £ 2 
a 2 b 2 c 2 . a 2 + A 6 2 + A c 2 — A ’ 
we take two points P x , P 2 on the first, and Q x , Q 2 the corresponding points on the 
second; then P x , P, being on a generating line of the first surface, Q x , Q 2 will be 
on a generating line of the second surface, and P X P 2 will be = Q X Q,. The same 
is evidently true for the quadrilaterals P X P 2 P 3 P 4 and Q X Q 2 Q 3 Q 4 , where P X P 2 , P 2 P 3 , 
P 3 P 4 , P 4 P X are generating lines on the first surface: and therefore Q X Q 2 , Q 2 Q S , Q 3 Q 4 , 
Q 4 Q x are generating lines on the second surface, which proves the theorem. 
9—2