787] 
FUNCTION. 
541 
return each of them to its original position. And it is easy to understand how, 
when the oval described by P surrounds two or more of the branch-points V, the 
corresponding curves for P' may be a system of manifold ovals, such that the sum of 
the manifoldness is always = n, and to conceive in a general way the behaviour of 
the corresponding points P and P'. 
Writing for a moment x + iy = u, x' + iy' — v, the branch-points are the points of 
contact of parallel tangents to the curve <f> (u, v) = 0, a line through a cusp (but 
not a line through a node), being reckoned as a tangent; that is, if this be a curve 
of the order n and class m, with S nodes and k cusps, the number of branch-points 
is = m 4- k, that is, it is = n 2 — n — 2$ — 2k, or if p, = ^ (n — 1) (n — 2) — 8 — k, be the 
deficiency, then the number is = 2n — 2 + 2p. 
To illustrate the theory of the w-valued algebraical function x' + iy of the complex 
variable x + iy, Riemann introduces the notion of a surface composed of n coincident 
planes or sheets, such that the transition from one sheet to another is made at the 
branch-points, and that the n sheets form together a multiply-connected surface, which 
can be by cross-cuts (Querschnitte) converted into a simply-connected surface; the 
n-valued function x + iy' becomes thus a one-valued function of x -f iy, considered as 
belonging to a point on some determinate sheet of the surface: and upon such con 
siderations he founds the whole theory of the functions which arise from the integration 
of the differential expressions depending on the w-valued algebraical function (that is, 
any irrational algebraical function whatever) of the independent variable, establishing 
as part of the investigation the theory of the multiple ^-functions. But it would be 
difficult to give a further account of these investigations. 
The Calculus of Functions. 
22. The so-called Calculus of Functions, as considered chiefly by Herschel and 
Babbage and De Morgan, is not so much a theory of functions as a theory of the 
solution of functional equations; or, as perhaps should rather be said, the solution of 
functional equations by means of known functions, or symbols,—the epithet known 
being here used in reference to the actual state of analysis. Thus for a functional 
equation (j>x + (fry = <fr (xy), taking the logarithm as a known function, the solution is 
(frx — c log x; or if the logarithm is not taken to be a known function, then a solution 
[dec 
may be obtained by means of the sign of integration (frx = c I —; but the establish- 
j ec 
ment of the properties of the function logarithm (assumed to be previously unknown) 
would not be considered as coming within the theory. A class of equations specially 
considered is where ax, ¡3x, ... being given functions of x, the unknown function efr is 
to be determined by means of a given relation between x, (frx, (frax, (fr/3x, ...; in part 
icular the given relation may be between x, (frx, (frax; this can be at once reduced 
to equations of finite differences; for writing x = u n , ax = u n+1 , we have u n+1 = au n , 
giving u n , and therefore also x, each of them as a function of n; and then writing 
<frx = v n , (frax will be the same function of n+1, = v n+1 , and the given relation is 
again an equation of finite differences in v n+1 , v n , and n; we have thus v n , = (frx,