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ANALYTICAL THEOREM RELATING TO SECTIONS &C. 
[332 
1. The sections x = Q, x'd2a + y'^2b+z*Jc + w*fd = 0 of the quadric surface 
aaf + by 2 + Qxy \/ab — cz 2 — dw 2 = 0 will touch each other if, combining together the 
equations 
x = 0, y s/2b + z Vc + w ^d = 0, by 2 — cz 1 — dw 2 = 0, 
these give a twofold value (pair of equal values) for the ratios y : z : w. We in 
fact have 
by 2 — cz 2 — dw 2 = by 2 — cz 2 — {y V 2b + z Vc) 2 , 
= — by 2 — 2cz 2 — 2yz V26c, 
= — (2/ ^b + z V2c) 2 ; 
and the right-hand side being a perfect square, the condition of contact is satisfied. 
2. In like manner we have the system 
z = 0, x^/2a + y*l2b + w Vd, = 0, ax? + by 2 + Qxy \/ab — dw 2 = 0, 
which gives 
aa: 2 + by 2 + Qxy \/ab — dw 2 
= a« 2 4- by 2 + 6aa/ VaZ) — (¿c V2a + y V26) 2 , 
= — ax? — &?/ 2 + 2#i/ Vab, 
= — (x\/a — y V6) 2 ; 
and here also, the right-hand side being a perfect square, the condition of contact is 
satisfied. 
Cambridge, November 28, 1863.