646 REPORT—1899. 
differences (m and being arbitrary integers, positive, negative, or zero), we are 
carried to any other. A"S~" carries us ” steps down a column, 6"A —” carries us 
n steps up it, and A”6" carries us 2n columns to the right. A” carries us steps 
obliquely down, and 6” the same number obliquely up, both to the right. Reversal 
of the sign of m reverses the direction. 
In the old notation, the fundamental property « 
Ad=A-—5d 
would be written 
A°’Yn-1 = AYn i: AYn-1 35 
which does not afford the same facility for manipulation. 
6. On the Partial Differential Equation of the Second Order. 
By Prof. A. C. Dixon. 
Taking the equation 
F(@; Yy %y P; q r, 8; t) =0, 
I suppose it solved by using two more relations, 
u=a, v=b, 
among the quantities 2, y, 2, p, 9,7, 8, t, to give values‘of r, s, t, which, substituted 
in 
dz=pdx + qgdy, dp=rdx + sdy, dq =sdx + tdy, 
render these three equations integrable. This will not be possible, of course, 
unless the expressions w, v fulfil certain conditions. I consider the case in which 
u can be so determined that v is only subjected to one condition, and I find that 
then du is a linear combination of the expressions 
dx + pdy, dz—pdx —qdy, dp —rdx—sdy, dq—sdx —tdy, 
Of 7, OF 4. Of OF Ok gihOF S , OF’ a. 
Pa at te am +p Oe +7 ap +5 4 Jan 
where p is a root of the quadratic 
of 24 Of Of _o, 
ott * as" Or 
These are the expressions used by Hamburger in his method of solution. : 
If such a function w can be found, the system f=0, w=a will have a series 
of solutions depending on an arbitrary function of one variable, and involving two 
further arbitrary constants. 
7. On the Fundamental Differential Equations of Geometry. 
By Dr. Irvine StRincHAM. 
Capitaine Feye Sainte-Marie, in his work ‘ Ktudes Analytiques sur la Théorie 
des Paralléles,’ after showing that the propositions of the Euclidean Geometry are 
true within an infinitesimal domain, achieves, through the processes of integration, 
a series of analytical formule for non-Euclidian geometry. ; ; 
The foundation for this analytical theory is a group of differential equations. 
