Arthur Cayley, John Purser, and Frederick Purser

“ A complex is a triply infinite system of lines-for instance, the tangent lines of a surface. Considering an arbitrary point in connexion with the complex, the complex lines which C I pass through the point form a cone; considering a plane amp Mies in connexion with it, the. complex lines which lie in the plane envelop a curve. It is 'easy to see that the class of the curve is equal to the order of the cone; in fact each of these numbers is equal to the number of complex lines which lie in an arbitrary plane and pass through an arbitrary point of that plane ”

“ A helicoidal surface means that traced out by a rigid wire, which is given a screw motion round a fixed axis, or, which comes to the same thing, the surface made up of a system of helices starting from the points of a given curve, all having the same axis and the same interval between the successive threads. ”

“ As explained above, the lines of curvature on a quadric are the systems of orthogonal curves formed by its intersection with the two systems of confocal quadrics. Ioachimsthal showed that the interpretation of the first integral of the equation for geodetics on a central quadric is, that along a geodetic pD =constant (C,) p denoting the perpendicular let fall from the centre on the tangent plane, and D the semidiameter drawn parallel to the element of the geodetic, the envelope of all geodetics having the same C being a line of curvature. ”

“ The normals of a. surface form a congruence. In any congruence the lines consecutive to a~ given congruence line do not in general meet this line; but there is a determinate number of consecutive lines which do meet it; or, attending for the moment to only one of these, say the congruence line is met by a consecutive congruence-line. In particular, each normal is met by a consecutive normal; this again is met by a consecutive normal, and so on. That is, we ha'e a singly infinite system of normals each meeting the consecutive normal, and so forming a torse ”

“ If curvature of C map circle C' intersects the base, the coaxal system cutting C' orthogonally does not intersect in a real point, and C has accordingly no centre. It is of interest to examine in what way a pseudo sphere differs from a plane as regards the behaviour of parallel lines. ”

“ For since the indicatrix at every point is a rectangular hyperbola, the angle between the elements of two intersecting curves-=angle between their conjugate tangents; but this =angle between conjugate tangents to representative curves on sphere=angle between these curves themselves.
The problem of finding a minimal surface to pass through a given curve in space, known as Plateau's problem, possesses an exceptional interest from the circumstance that it can be always exhibited to the eye in the following way by an actual physical experiment. ”

“ The expression for the specific curvature, which is in general somewhat elaborate, assumes a very simple form when a system of geodetics and the system of their orthogonal trajectories are chosen for the parameter curves, the parameter u being made the length of the arc of the geodetic, measured from the curve, u=o selected as the standard. ”

“ If at each point of the given surface we measure off along the normal one and the same distance at pleasure, then the locus of the points thus obtained is a surface cutting all the normals of the given surface at right an les, or, in other words, having the same normals as the given surface ”

“ A torse or develop able corresponds to a curve in space in the same manner as a cone corresponds to a plane curve: although capable of representation by an equation U = (* § x, y, z, w) " =o, and so of coming under the foregoing pointfrorse °' I definition of a surface, it is an entirely distinct geo-Devempah e metrical conception. We may indeed, qua surface, regard it as a surface characterized by the property that each of its tangent planes touches it, not at a single point, but along a line ”

“ There seems to be a prevalent opinion amongst mathematicians that such deformation is always impossible, but we do not think any unimpeachable demonstration of this has yet been given. It is certain that a complete spherical surface does not admit of in extensive deformation, for if it did it would follow from Gauss's theorem that the new surface would have a uniform specific curvature. ”

“ The complete enumeration and discussion of the singularities of a surface is a question of extreme difficulty which has 'not yet been solved! A plane curve has.point singularities and'line singularities; corresponding to these we have for the surface isolated point singularities and isolated plane singularities, but there are besides continuous singularities applying to $i"g"""i curves on or torses circumscribed to the surface, 'and nes it'is 'among these that we, have the non-special singularities which play the most important part in the theory. ”

“ Geddelics on Developable Surfaces.-On these the geodetics are the curves which become right lines when the surface is unrolled into a plane. From this property a first integral can be immediately deduced. - 3. Geodetics on Surfaces of Revolution.-In all such the geodetics are the curves given by the equation r sin ¢=const, 1' being the perpendicular on the axis of revolution, d> the angle at which the curve crosses the meridian. The general problem of the determination of geodetics on any surface may be advantageously treated in connexion with that of “ parallel " curves. ”

“ Chasles proved that if an ellipse be intersected in the point A by a. confocal hyperbola, and from any point P on the hyperbola tangents PT, PT' be drawn to the ellipse, then the difference of the arcs of the ellipse TA, T'A=the difference of the tangents PT, PT'; and subsequently Graves showed that if from any point P on the outer of two confocal ellipses tangents be drawn to the inner, then the excess of the sum of the tangents PT, PT' over the intercepted arc TT' is constant. ”

“ It is noteworthy that the expression for the lineal element on a minimal surface assumes the isothermal form ds2=> (du'-I-11122) - (I) when the curves u=const, v=const are so chosen as to be the lines of curvature; and (2) when they are chosen to be the lines in which the surface is intersected bly a system of parallel planes and the orthogonal trajectories of t ese lines. It is easily roved that a minimal surface possesses the property of being conflcirmable to its spherical representation. ”