428 [279 
279. 
ON A THEOREM RELATING TO SPHERICAL CONICS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. in. (1860), p. 53.] 
The following theorem was given by Prof. Maccullagh: “ If three lines at right 
angles to each other pass through a fixed point 0 so that two of them are confined 
to given planes: the third line traces out a cone of the second order whose sections 
parallel to the given planes are circles, and the plane containing the other two lines 
envelopes a cone of the second order whose sections by planes parallel to the given 
planes are parabolas.” 
Referring the figure to the sphere we have a trirectangular triangle XYZ, of which 
two angles X, Y lie on fixed arcs A, B. The angle Z generates a spherical conic U' 
having A, B for its cyclic arcs. The side XY envelopes a spherical conic U touched 
by the arcs A, B. The conic U' is evidently the supplementary conic of U, hence 
the poles of A, B are the foci of U. We may drop altogether the consideration of 
the triangle XYZ and consider only the side XY, we have then the theorem: 
If a quadrantal arc XY slides between the two fixed arcs A, B, the envelope of 
XY is a spherical conic U touched by the fixed arcs A, B, and which has for its 
foci the poles of these same arcs A, B. 
It is worth while to notice the great reduction of order which takes place in 
consequence of the arc XY being a quadrant. If XY had been an arc of a given 
magnitude 6, the envelope would have been a spherical curve of an order certainly 
higher than 6. For considering the corresponding problem in piano, the envelope in 
the particular case where the fixed lines A, B are at right angles to each other 
is a curve of the sixth order, and in the general case where the two fixed lines are 
not at right angles the order is higher: the problem in piano corresponds of course, 
not to the general problem on the sphere, but to that in which 6 is indefinitely small.