414 TRANSACTIONS OF SECTION A. 
(7) (c°)it is necessary and sufficient that the principal residue relative to the value 
z=co in the product R fe w) y (z, u) should vanish. Furthermore, on taking the 
integer i sufficiently large and equating to o, the principal residue in the product 
n t+1 Mos —i 
—7,ni— — — 
S BS Yop n—-127"U = Zee Wee Lyt—1, 
t=lr=-t tel r=i+1 
it can be shown that we impose on the second factor the orders of coincidence furnished 
by the partial basis ( 7 ) (co), We thus impose on the coefficients ar — 1, — 1in the second 
factor a number of conditions equal to the number of the coefficients y — r,n —2 in 
the first factor, which are arbitrary. Designating then by 2”(i-+ 1) — A and A respec- 
tively the number of arbitrary coefficients in the first factor and the number of co- 
efficients remaining arbitrary in the second factor after equating to zero the principal 
residue in the product, it is readily seen that we must have 
roo 1 mae : 
2n (t+ 1)-—2A= = (7 2% — #2) y ©), 
s=1 & & & 
We have, however, by hypothesis 
co 
‘co =, ] 
as D4 7 <2) = pO) 1+ 5 (cop 8 = ocean 
& 
We immediately derive 
‘ (©) y “9) a ‘co) 1, ‘co) 
= , . —-7F : _ —N4 
A=ni+ ni ape es be {Hs 1+ Foes s 
This expression for A then gives us the number of the independent conditions which 
must be imposed on the coefficients of the general rational function R ¢ , u) of degree 
7 in z in order that it may have for the value z = co the orders of coincidence furnished 
by the partial basis (r)(co)—the integer i being here taken not less than the highest 
degree in z which a rational function of (z, w) can have and yet possess the orders of 
coincidence here in question. The corresponding result for any finite value z=a 
can be immediately stated. 
5. The Use of the Exponential Curve in Graphics. 
By Dr. H. B. Heywoop. 
The properties of the exponential curve which are employed are shown in 
the following table, together with the graphical operations to which they may 
be applied :— 
Error. 
—_4—_—_ 
Maximum. Average, 
Per cent. Per cent. 
Ls net ee Hier 2 (Multiplication) ae Ss 1 06 
Die etc fe! b= ete” (Division) ... Ae oe 1 — 
oes) ea 634 (Evolution, &c. AG aes 3 0-3 
4, AO =e (Differentiation) ab ws (15) 5 
b 
5. j e*dx =e’ —e* (Integration) ... A i 1 0°6 
For carrying out the processes we use a templet of transparent celluloid, upon 
which is marked a graduated exponential curve. 
To perform the multiplication of two numbers, y, X y,, we find with a pair 
of dividers the abscisse, x, and x,, which correspond to the ordinates y, and y,, 
The product is the ordinate corresponding to the abscissa x, + 2,. F 
The second and third operations are carried out on similar lines. 
