20 REPORT—1904. 
of the kites and apparatus. With regard to the latter an easy means 
of calibrating the meteorograph is required, and this involves the use of 
a suitable air-tight inclosure which might be used for similar operations 
in future. The cost of such an apparatus is estimated at 25/. 
The Committee therefore ask for reappointment, with a grant of 401. 
Report on the Theory of Point-groups.'—Parr IV. 
By Frances Harpeastie, Cambridge. 
§ 10. 1857-18783. ‘The rise and development of the theory of algebraic 
functions has been described by Brill and Noether in a masterpiece of 
German erudition.2 The whole of the fifth and sixth sections of this 
work, as well as large portions of other sections, treat, historically and 
critically, of matters more or less relevant to the theory of point-groups. 
It would be impracticable to deal thoroughly with such a mass of 
material within the limits of this Report. Some account of certain 
publications of this period will, however, indicate how the theory of 
functions came to be connected with the theory of higher plane curves, 
a connection which is the origin of the theory of point-groups. And as 
a preliminary to this, the following section deals very briefly with 
Riemann’s contribution to the ideas which gave rise to this theory. 
At the very end of the eighteenth century, two years before the 
birth of Pliicker and five years before that of Jacobi, Gauss, then in 
his twenty-second year, put forward a strictly rigorous proof of the 
fundamental theorem of algebra. In this dissertation* the position of 
a point in the plane was taken, for the first time, as the geometrical 
interpretation of a single complex variable, in contrast to the Cartesian 
plan, which, confining the attention to real quantities, had associated 
two independent variables with each point. By this interpretation the 
first step was taken towards the connection of the theory of functions 
with the theory of higher plane curves ; for, fifty-two years later, this 
plane of the complex variable was made the foundation of the ingenious 
structure commonly known as a ‘ Riemann surface.’ The conception of 
a many-sheeted surface spread over the plane, upon which the complex 
variable is free to move, and in which the sheets are connected by inter- 
penetration of one another in a manner which it is impossible to con- 
struct in the concrete, but which in the imagination affords a perfect 
representation of the ‘branching’ of a many-valued function, each sheet 
being associated with one branch of the function, and melting into 
another sheet round a point at which, for one and the same value of the 
variable, two branches coincide—this conception, by means of which the 
many-valued function is transformed into a one-valued function of the 
position of the variable on the surface, is due to Riemann (1826-1866), 
who first described such a surface in his ‘Inaugural Dissertation,’ 
! Parts I., II., and III. appeared in the Brit. Assoc. Reports for 1900, 1902, 1903. 
2 Brill and Noether, ‘ Die Entwicklung der Theorie der algebraischen Functionen 
in iilterer und neuerer Zeit’ (Jahresber. d. deutschen Math. Ver. vol. iii. (1894), 
pp. 109-565). 
3 *Demonstratio nova theorematis, onnem functionem algebraicam rationalem 
integram unius variabilis in factores reales primi vel secundi gradus resolvi posse,’ 
Helmsiadt, 1799 ; Gauss, Werke, vol. iii, 1876, pp. 1-30. 
