41 
of Edinburgh, Session 1872-73. 
ing on the determination of the constants in each finite integral 
so as to satisfy the limiting conditions of the problem. To a few 
terms we have 
| x + n 
| x- 1 | n+ 
- j 2 + (2rc-l)lL±- 1 + {n- 2) 2 (- + 1)n . 
1 \ x x(x + 1) ) 
By a slight modification of the preceding process we get in 
succession 
A? — x, x + n — p — (x + 1), x-\-n + P— 
(x + 2) , X + n , 
■X, X + 
I g+J: U I /- I'^-I , 1)(»-2)(»- 1) , 
[ x f 1 \ n - 1 ( v , x + 2 2 (x + 2)(x + Z) 
The graphical method to which I referred above consists simply 
in supposing the various values of F x> y to be written each at the 
point whose co-ordinates are the values of a? and y. If, to fix the 
ideas, we suppose the axis of x to be horizontal and that of y 
vertically downwards, then the fundamental equation shows that 
by adding together any three contiguous numbers in a horizontal line , 
we 'produce the number immediately under the middle one of the three. 
The limiting conditions show that all the numbers along the 
line 
x + y = 0, 
and those between it and the negative part of the axis of x , are 
zeros ; while those along 
y = x - 1 , y = x - 2 , y + x = 1 , 
are each equal to 1 . 
Hence we have the figure 
0000001100 0 0....X 
00000 1211000 
00001 3441100 
00014 8 11 96110 
0 0 1 5 13 23 28 26 16 8 1 1 
0 1 6 19 41 64 77 70 50 25 10 1 
&c. y &c. 
• («) 
