120 
Professor Challis said that though impossible quantities 
might not be very good instruments of education, they were, 
in higher investigations, absolutely necessary, and true in the 
strictest sense. 
Professor Cayley believing that the theory of impossible 
quantities was absolutely true, did not wish it to be excluded 
from the University studies or placed in a secondary position. 
Professor Adams, admitting that imaginary quantities might 
be used without proper logic, considered that though un- 
discovered there must be a logic discoverable, and thought the 
Astronomer Royal’s proof of the Theorem not the natural proof, 
since it was so complicated. He also held that the proof of 
imaginary quantities could be, and, in some cases had been 
rendered strictly logical. 
(2) On some Porismatic Problems. By W. K. CLIFFORD. 
The PRosLeM :—To draw a polygon of a given number of 
sides, all whose vertices shall lie on one given conic, and all 
whose sides shall touch another given conic: is either not pos- 
sible at all, or possible in an infinity of ways. This remark, 
originally made by Poncelet, has been shewn by Professor 
Cayley to depend in a very beautiful manner upon the theory 
of elliptic functions; and in this way he has proved that an 
analogous theorem holds good wherever a (2, 2) correspondence 
exists: that is to say, whenever two things are so related that 
to every position of either there correspond two positions of the 
other. ‘Two points a, y for instance, in a conic U, which are 
connected by the relation that the line «y touches a second 
conic, V, have a correspondence of this kind: for if the point x 
be taken arbitrarily, two tangents cana be drawn from it to V, 
determining two positions of y: and conversely, the point y 
being fixed determines two positions of #. The theorem is then 
that in a (2, 2) correspondence there is either no closed cycle 
of a given order, or an infinite number. In the present com- 
