28 REPORT—1883. 
equations ; it is not every curve, but only a curve which is the complete 
intersection of two surfaces, which can be properly represented by two 
equations (2, y; z, w)”" = O, (a, y, 2, w)” = O, in quadriplanar coordinates ; 
and in regard to this question, which may also be regarded as that of the 
classification of curves in space, we have quite recently three elaborate 
memoirs by Nother, Halphen, and Valentiner respectively. 
In n-dimensional geometry, only isolated questions have been con- 
sidered. The field is simply too wide; the comparison with each other 
of the two cases of plane geometry and solid geometry is enough to show 
how the complexity and difficulty of the theory would increase with each 
successive dimension. 
In Transcendental Analysis, or the Theory of Functions, we have all 
that has been done in the present century with regard to the general 
theory of the function of an imaginary variable by Gauss, Cauchy, 
Puiseux, Briot, Bouquet, Liouville, Riemann, Fuchs, Weierstrass, and 
others. The fundamental idea of the geometrical representation of an 
imaginary variable «+ iy, by means of the point having «, y for its 
coordinates, belongs, as I mentioned, to Gauss; of this I have already 
spoken at some length. The notion has been applied to differential 
equations ; in the modern point of view, the problem in regard to a 
given differential equation is, not so much to reduce the differential 
equation to quadratures, as to determine from it directly the course of the 
integrals for all positions of the point representing the independent 
variable: in particular, the differential equation of the second order 
leading to the hypergeometric series F(a, 3, y, x) has been treated in 
this manner, with the most interesting results; the function so deter- 
mined for all values of the parameters (a, 3, y) is thus becoming a known 
function. I would here also refer to the new notion in this part of 
analysis introduced by Weierstrass—that of the one-valued integer func- 
tion, as defined by an infinite series of ascending powers, convergent for 
all finite values, real or imaginary, of the variable « or 1/#— cc, and so 
having the one essential singular point «=o or «=c, as the case may 
be: the memoir is published in the Berlin Abhandlangen, 1876. 
But it isnot only general theory : I have to speak of the various special 
functions to which the theory has been applied, or say the various known 
functions. 
For a long time the only known transcendental functions were the 
circular functions sine, cosine, &c.; the logarithm—v.e. for analytical 
purposes the hyperbolic logarithm to the base e; and, as implied therein, 
the exponential function e*. More completely stated, the group comprises 
the direct circular functions sin, cos, &c.; the inverse circular functions 
sin“! or aresin, &c.; the exponential function, exp.; and the inverse 
exponential, or logarithmic, function, log. 
Passing over the very important Eulerian integral of the second 
kind or gamma-function, the theory of which has quite recently given 
