586 
NOTES AND REFERENCES. 
20. The discovery of the formula q (ix +jy + kz) q~~ l = ix' +jy' + kz', as expressing 
a rotation, was made by Sir W. R Hamilton some months previous to the date of 
this paper. As appears by the paper itself, I was led to it by Rodrigues’ formulse, 
see 6. For the further development of the theory, see 68. 
21. The system of imaginaries i lf i 2 ,...i 7 had presented itself to J. T. Graves about 
Christmas 1843, see his paper “ On a Connection &c.” Phil. Mag. vol. xxvi. (1845), 
pp. 315—320. They are called by him Octads, or Octaves. 
24 and 25. These papers precede the researches of Eisenstein on the same subject. 
Grelle, t. xxxv. (IS! 1 ?). It was I think right that the theory of the doubly infinite 
products should have been investigated as in these papers: but the investigation is in 
some measure superseded by the beautiful theory of Weierstrass, viz. he takes for the 
element of the product (not a mere linear function of u, but) a linear function multiplied 
by an exponential factor, 
au = ¿HI 
w = 2to<m + 2m'w' where the ratio w : w' is imaginary, and the product extends to all 
positive or negative integer values of m, m! (the simultaneous values 0, 0 excluded) : 
in consequence of the introduction of the exponential factor the form of the bounding 
curve becomes immaterial, and the only condition is that it shall be ultimately every 
where at an infinite distance from the origin. 
The general theory of Weierstrass in regard to the exponential factor is given in 
the Memoir, “ Zur Theorie der eindeutigen analytischen Functionen,” Berlin. Abh. 1876 
(reprinted, Abhandlungen aus der Functionenlehre, 8° Berlin, 1886): and the application 
to Elliptic Functions is made in his lectures, edited by Schwarz, Formeln und Lehrsätze 
u. s. w. 4° Gott. 1883. See also Halphen, Théorie des Fonctions Elliptiques, Paris, 1887. 
26. The geometrical results in regard to corresponding points on a cubic curve 
are many of them due to Maclaurin. See his “ De linearum geometricarum proprieta 
tibus generalibus tractatus,” published as an Appendix to his Treatise on Algebra, 
5 Ed. Lond. 1788. (See also De Jonquières’ “Melanges de Géométrie pure,” 8° Paris, 
1856.) But the theorem in the “Addition” was probably new: the curve of the third 
class touched by the line PP' is the curve called by me the Pippian (as represented 
by the contravariant equation PJJ = 0), but which has since been called the Cayleyan 
of the cubic curve. 
27. The theory of the conics of involution was so far as I am aware new. 
28 and 29. See 2, 3. 
30. As noticed in the paper, the investigation is directly founded upon that of 
Pliicker for the singularities of a plane curve. It is to be observed that the definition 
of a “ line through two points ” ligne menée par deux points (non consecutifs en général) 
du système, does not exclude actual double points, for the line through an actual 
double point is a line through two points, coincident indeed, but not consecutive; but