20 
NOTICES OF COMMUNICATIONS TO THE 
[387 
algebraical sum of the distances of a point thereof from three given foci is = 0 (this 
was selected for facility of construction, by the intersections of circles and confocal 
conics). The quartic consists of two equal and symmetrically situated pear-shaped 
curves, exterior to each other, and including the one of them two of the three given 
foci, the other of them the third given focus, and a fourth focus lying in a circle 
with the given foci: by inversion in regard to a circle having its centre at a focus 
the two pear-shaped curves became respectively the exterior and the interior ovals of 
a Cartesian. There was also a figure of the two circular cubics, having for foci four 
given points on a circle; and a figure (coloured in regions) in preparation for the 
construction of the analogous sextic curve derived from four given points not in a circle. 
March 28, 1867. pp. 25—26. 
Professor Cayley mentioned a theorem included in Prof. Sylvester’s theory of 
derivation of the points of a cubic curve. Writing down the series of numbers 
1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, &c., viz., all the numbers not divisible by 3, 
then (repetitions of the same number being permissible) taking any two numbers of 
the series, we have in the series a third number, which is the sum or else the 
difference of the two numbers (for example, 2, 2 give their sum 4, but 2, 7 give their 
difference 5), and we have thus a series of triads, in each of which one number is 
the sum of the other two. The theorem is, that it is possible on a cubic curve to 
construct a series of points, such that denoting them by the above numbers respectively, 
then for any triad of numbers as aforesaid the points denoted by the three numbers 
respectively lie in lined. And the theorem gives its own construction: in fact the 
series of triads is 112, 224, 145, 257, 178, 248, &c. Take 1, an arbitrary point on the 
cubic, then (by the theorem) the triad 112 shows that 2 is the tangential of 1; 
224 shows that 4 is the tangential of 2; 145 that 5 is the third point of 1 and 4; 
257 that 7 is the third point of 2 and 5. So far we have no theorem; we have 
merely, starting from the point 1, constructed by an arbitrary process the points 
2, 4, 5, and 7. But going a step further; 178 and 448 show, the first of them, that 
8 is the third point of 1 and 7, the second of them, that 8 is the tangential of 4. 
We have here the theorem that the third point of 1 and 7 is also the tangential 
of 4. Similarly, 10, 11, 13 are each of them (like 8) determined by two constructions; 
14, 16, 17, 19, each of them by three constructions, and so on; the number of con 
structions increasing by unity for each group of four numbers. And the theorem is, 
that these constructions, 2, 3, or more, as the case may be, give always one and the 
same point. Prof. Cayley mentioned that on a large figure of a cubic curve he had, 
in accordance with the theorem, constructed the series of points 1, 2, 4, 5, 7, 8, 10, 11, 
13, 14. 
April 15, 1867. p. 29. 
Prof. Cayley communicated a theorem relating to the locus of the ninth of the 
points of intersection of two cubics, seven of these points being fixed, while the eighth 
moves on a straight line.