4 REPORT—1845. 
If now about a point we describe a circle with radius a, and also about the same 
point trace the curve 
e=at {gq cosnb+q,cosm—1O0+......+%}, ear C=) 
and read off the values of 4 when this curve cuts the circle, 7. e. when e = a, we thus 
get all the values of 4 that satisfy the equation, 
J cosnd+q,cosm—16+...... +4, =0, 
or which satisfy (6.); and taking the cosines of these angles, we get the values of a 
not beyond the limits + 1 and — 1 which satisfy «. 
To find the remainder of the real roots of (#.), we may write - for x, and find the 
values of y between + 1 and — 1; and inverting these, we find the values of x which 
satisfy («.) that lie beyond the limits + 1 and — 1. 
This solution, in fact, depends on a ready method of trying all values of 2 between 
+ 1 and — 1, and selecting those which satisfy the equation. 
It may be observed, that 
as @ changes from 0 to x, « or cos 6 changes from + 1 to —1; 
ANd!as 8) ~ jenes )ieke Vato) Vary v@Or'CO8O) wes ¢ eesy = Litoteeia 
and therefore it is evident, from the form of (6.), that the curve (0.) will have an axis, 
and for every root of (#.) we shall have two readings. If the readings give the same 
value of a, this will probably be the true value ; if different, they will most probably 
be limits, as in one case 2 is in the course of being gradually diminished from + 1 
to — 1, and in the other of being increased from —1 to +1. 
We have also a means of estimating the probable correctness of a root furnished 
by the several angles at which the curve cuts the circle. _ ‘By the help of this engine 
we may also trace curves of the form 
e= 2b cos m {cos [cos (r+ «) + a! + al} 
On a System of Numerical Notation. By Tuomas Wricat Hitt. 
This was proposed to be founded upon the number 16, and those derived from it 
by successive division by 2,—such as 8, 4, 2,1. By the combinations of these all 
numbers were to be formed, and by attaching letters as the marks or names for the 
elementary numbers, a system of nomenclature was obtained. The plan also pro- 
vided figures and names for them essentially negative, and thus fit for promiscuous 
incorporation with positive numerals on whatever scale of notation. 
On the Nebula 25 Herschel, or 61 of Messier’s Catalogue. 
By the Eanrt or Rosse. 
Lord Rosse exhibited to the Section what he called his working plan of this nebula, 
and explained his method. He first laid down, by an accurate scale, the great fea- 
tures of the nebula as seen in his smallest telescope, which, being mounted equa- 
torially, enabled him to take accurate measurements ; he then filled in the other 
parts, which could not be distinguished in that telescope, by the aid of the great 
telescope; but as the equatorial mounting of this latter was not yet complete, he 
could not lay these smaller portions down with rigorous accuracy; yet as he had 
repeatedly gone over them, and verified them with much care, though by estimation, 
he did not think the drawing would be found to need much future correction. 
———_--——- 
Sir J. Herschel exhibited a model of the globe of the moon in relief, expressing 
the forms and elevations of its mountains as seen in a good telescope. This beau- 
tiful and exquisite work he stated to be the performance of a Hanoverian lady, 
Madame Witte; modelled by her from actual observation through an excellent 
Fraunhofer telescope, i in a small observatory at the top of her own dwelling-house ; 
the selenographical positions and general contours of the principal craters and other 
