A MEMOIR ON THE ABELIAN AND THETA FUNCTIONS. 
[Chapters I to III, American Journal of Mathematics, t. v. (1882), pp. 187—179; 
Chapters IV to VII, ib., t. vn. (1885), pp. 101—167.] 
The present memoir is based upon Clebsch and Gordan’s Tlieorie der Abel’schen 
Functionen, Leipzig, 1866 (here cited as C. and G.); the employment of differential 
rather than of integral equations is a novelty; but the chief addition to the theory 
consists in the determination which I have made for the cubic curve, and also (but 
not as yet in a perfect form) for the quartic curve, of the differential expression dTi^ v 
[fi 
(or as I write it dII 12 ) in the integral of the third kind dll^ in the final normal 
J a 
form (endliche Normalform) for which we have (p. 117) j c2II a/3 = I dJJ^ v , the limits 
and parametric points interchangeable. The want of this determination presented itself 
to me as a lacuna in the theory during the course of lectures on the subject which 
I had the pleasure of giving at the Johns Hopkins University, Baltimore, U.S.A., in 
the months January to June, 1882, and I succeeded in effecting it for the cubic curve; 
but it was not until shortly after my return to England that I was able partially 
to effect the like determination in the far more difficult case of the quartic curve. 
The memoir contains, with additional developments, a reproduction of the course of 
lectures just referred to. I have endeavoured to simplify as much as possible the 
notations and demonstrations of Clebsch and Gordan’s admirable treatise; to bring 
some of the geometrical results into greater prominence; and to illustrate the theory 
by examples in regard to the cubic, the nodal quartic, and the general quartic curves 
respectively. The various chapters are: I, Abel’s Theorem; II, Proof of Abel’s Theorem; 
III, The Major Function; IV, The Major Function (continued); V, Miscellaneous 
Investigations; VI, The Nodal Quartic; VII, The Functions T, U, V, ©. The 
paragraphs of the whole memoir will be numbered continuously.