ON THE TRANSFORMATION OF TWO By WILLIAM WALTON, M.A., Fellow of Trinity Hall. THE object of this article is to prove that the two simultaneous equations can be satisfied by no other independent system. Multiply the equations (3), (4), and the identical equation by λ, μ, v, respectively, add together the resulting equations, and equate to zero the coefficients of 1 1 1 v w and we have v) = 0; u (b −c) + v (c − a) + w (a − b) = 0....................... (5). The equation (3) may be written in the form ..... and the equations (5), (6), respectively by 1, A, u, adding, and equating to zero the coefficients of u, v, w, we have Adding together the equations (7), we have or au (b−c) + bv (c− a) + cw (a - b) = 0.................... (8). Multiplying the equations (7) by a, b, c, respectively, and adding, we get, attending to (5), The equations (8) and (9) are not to be taken simultaneously, but alternatively. In fact, if we combine them, we shall easily ascertain that relations which do not satisfy the equation (5). Combining (5) and (8), we have at once Putting for u, v, w, their values, the systems (10) and (11) become respectively Thus we see that these two sets of relations, either of which obviously satisfies the simultaneous equations (1) and (2), are the only independent systems which can do so. The preceding investigations are equivalent to proving the converse of the proposition— "that the intersection of a quadric with a confocal quadric is a common line of curvature of the two quadrics ;"* ̄`viz.— Mr. Thomson (now Professor Sir William Thomson); Cambridge Mathematical Journal, Vol, Iv., p. 283. "that if two concentric and similarly situated quadrics intersect in a common line of curvature, they are confocal.” Let, in fact, the equations to two confocal quadrics be intersects (14) in the line of curvature of (14) in which (14) Eliminating and 7 between the equations (17) we shall easily ascertain that as the condition that the intersection of (14) and (16) shall be a line of curvature of (14). Similarly the condition that the intersection of (14) and (16) shall be a line of curvature of (16) is a (B-y) B(y-a) y (α-B) + a a b-B + = = 0. Of the systems (12) and (13), deduced above from these two equations, the former shews that (14) and (16) are confocal; the latter shews that (14) and (16) are either coincident surfaces or that they do not intersect. Observe, however, as regards the system of the three solutions, that if for instance the quadric surfaces are these surfaces are not confocal but they intersect (touch) in the ellipse z=0, x2 y2 + =1, which is a principal section, and a b consequently a curve of curvature, of each of them; and similarly for the other two of the three solutions. Note by Prof. Cayley. Writing in Mr. Walton's equations (1) and (2) a b с а в Y a' d' d' 8' 8' 8 instead of a, b, c, a, ß, y respectively; and putting for shortness whence also aAF+bBG+cCH=0)...............(18), aAF+BBG +уCH=0; Now regarding (a, B, y, 8) as the coordinates of a point in space, the equations (18) and (19) represent each of them a cone having for vertex the point a: By: Sab:c:d, |