344 Mr J. Brill, On Solutions of Differential [May 6, 
a + vy =sn (a' + vy’) this establishes a (1, 1) correspondence between 
the wy infinite quarter plane and the 2’ y' rectangle (sides AK and 
i’): it 1s shown how to the contour A’B’C'D’ (A’ the origin, B’, 
D' on the axes of a’, y’ respectively) there corresponds the contour 
ABCD of the infinite quarter plane (A the origin, B at the 
distance 1 and C at the distance yon the axis of x, D at infinity, 
that is extending from infinity on the axis of # to infinity on the 
axis of y): and this shows at once the general form of the curve 
in the wy quarter plane corresponding to any given curve in the 
a'y' rectangle; for instance, to a straight line #’F” parallel to the 
axis of w and extending from #’ on A’D' to F’ on BC’ there cor- 
responds an arc HF extending from # on AD to F on BC, and 
cutting each of these lines at right angles: and so im other cases. 
The curves thus corresponding to straight lines HF" and G’H 
parallel to the axes of « and y' respectively are as is known 
Bicircular Quartics, and reference is made to an interesting paper 
by Siebeck, Crelle, t. 57 (1860). and t. 59 (1861): although the two 
theories are substantially identical, some of the properties are in 
the first instance developed in regard to the binodal quartic, and 
the term binodal quartic is accordingly introduced into the title 
of the Memoir. The bicircular quartics arising from the elliptic 
functions are biaxal curves, represented by an equation of the form 
(a + y’)? — 2Aa’ — 2By’ + C=0. 
(2) A Method of discovering Particular Solutions of Certain 
Differential Equations, that satisfy Specified Boundary Conditions. 
By J. Britt, M.A., St. John’s College. 
1. Riemann has given a method* by means of which we can 
discover solutions of a particular type of linear partial differential 
equations of the second order, with two independent variables, 
which give a specified value for the dependent variable along a 
given curve, as well as a specified value for its rate of variation 
in the direction of the normal to the curve. But in the majority 
of problems in Mathematical Physics, whose discussion involves 
linear equations of the second order, the boundary conditions are 
not of this character. We either have the value of the dependent 
variable given along each of two bounding curves, or we have its 
rate of variation in the direction of the normal given along each 
of the two curves. Or, speaking generally, instead of having two 
boundary conditions to satisfy along one given curve, we have one 
* See a Memoir entitled ‘‘Ueber die Fortpflanzung ebener Luftwellen von 
endlicher Schwingungsweite.” Gesammelte Werk, p. 145. A very lucid account. 
of Riemann’s method is given by Darboux, ‘“‘ Legons sur la Théorie Générale des 
Surfaces,” Livre rv., Ch. tv. 
