AI” REPORT—1883. 
then extend their definition unto the province of the real numbers, supposing each 
to be expressed in the binary scale (the negative numbers as ‘ arithmetical comple- 
ments’ with an infinity of l’s preceding). 
Then a+0 and a. 6 ought but to represent the results of combining together the 
homologous digits of a and 4 according to these tables. 
In order to avoid the particular fact that a+ would coincide with minus ad, 
let the positive part of the axis of the real numbers be reflected into itself accord- 
ing to any principle whatever, but such that the point 0 would correspond to 
itself. Do then the same with the negative part, but according to any other prin- 
ciple. Then the substitutes will evidently bear the same formal relations to each 
other as the original numbers, provided, of course, that we now define ad as the | 
point or number representative of the symbolical product, heretofore described, of 
those numbers whose representatives are a and 6 (and so on). 
Finally, the explanations may be extended to the ordinary complex numbers 
a+bxVJ/-1 
by simply combining together their real parts, and again the real coefficients of the 
imaginary parts, according to the rules given. 
10. Huposition of a Logical Principle, as disclosed by the Algebra of Logic, 
but overlooked by the Ancient Logicians. By Dr. Ernst ScHRODER. 
The algebra of logic being the concisest expression of the ‘laws of thought’ 
in the Boolean sense, enables us to discover gaps in the ancient system. 
By the method of Ch. Peirce the law of distribution 
a(b+c)=ab+ac 
is capable of being demonstrated through syllogisms, but merely as an implication 
in the one sense, viz., it can be shown that a( +c) zmplies ab + ac. 
The opposite implication has in a similar way never been proyed, and the impos- 
sibility of this proof being ever given may be demonstrated by means of a certain 
‘calculus with algorithms or calculusses,’ the notion of which is to be founded on 
the foreroing communication. 
This implication, therefore, is to be considered as an independent axiom of 
thought or logical principle. 
11. On Curves of the Fourth Class, with a Triple and a Single Focus. 
By Henry M. Jerrery, F.R.S. 
1. These ovals may be either singular or non-singular. They may be smooth, 
or have excrescences called stapetes, characterised by two cusps and a crunode. 
Let P, Q denote the triple and single forms of such a quartic; R, S the single 
foci of its satellite-conic; p, g, 7, s the perpendiculars drawn from them on any 
tangent. All such quartics have the property 
kp*qg=rs+X. 
Their equation, when referred to two-line tangential co-ordinates, is 
O= —x(1 +d) +mé+nn—1) (pEé+qn—1) (2 +97) 4+AE +7 P=od. 
Singular quartics are discriminated by a subsidiary curve (D,), which exhibits 
the mutual relation between the parameters x and A, when 
A diagram exhibiting this curve is a chart of ‘deficiency.’ 
