506 
ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
[869 
where n is IL, II3,... denote given functions of p, a, ft. Taking n odd and = 2s + 1 
we assume for y an expression 
_ x (Ag + A g _i x 2 + ... 4- M^ 2 * -2 + x 2s ) 
y 1 + A x x 2 + ... + A g ^xP~ 2 + A s cc 2s 
where the last coefficient A s is at once seen to be — p. Comparing with the series- 
value y = px(l + II 1 as 2 + IL& ,4 +...), we have an infinite series of equations. The first 
of these is, in fact, A s — p ; the next (s — 1) equations give linearly A 1} A 2 ,..., A s ^ 
in terms of the coefficients IT; that is, of p, a, ft: the two which follow serve in effect 
to determine p, ft as functions of a: and then, p and ft having these values, all the 
remaining equations will be satisfied identically. 
The process is an eminently practical one, so far as regards the determination 
of the coefficients A 1} A 2 ,..., A s ^ as functions of p, a, ft; it is less so, and requires 
eliminations more or less complicated, as regards the determination of the relations 
between p, a, ft. As to this, it may be remarked that the problem is not so much 
the determination of the equation between p and a (or say the pa-multiplier equation, 
or simply the pa-equation), and of the equation between ft, a (or say the a/3-modular 
equation, or simply the adequation), as it is to determine the complete system of 
relations between p, a, ft; treating these as coordinates, we have what may be called the 
multiplier-modular-curve, or say the MM-curve, and the relations in question are those 
which determine this curve. 
In the absence of special exceptions, it follows from general principles that the 
coefficients A 1} A 2 ,..., A s _ lt qua rational functions of p, a, ft, must also be rational 
functions of a, ft or of a, p; and I think it may be assumed that this is the case; 
the method, however, affords but little assistance towards thus expressing them. 
In connexion with the foregoing theory, I consider the solutions of the problem 
of transformation given by Jacobi’s partial differential equation (“Suite de Notices sur 
les Fonctions elliptiques,” Crelle. t. iv. (1829), pp. 185—193), and by what I call the 
Jacobi-Brioschi differential equations. The first and third of these were obtained by 
Jacobi in the memoir*, “De functionibus ellipticis Commentatio,” Crelle, t. iv. (1829), 
pp. 371—390 (see p. 376); but the second equation, which completes the system, was, 
I believe, first given by Brioschi in the second appendix to his translation of my 
Elliptic Functions: Trattato elementare delle Funzioni ellittiche: Milan, 1880. I had, 
strangely enough, overlooked the great importance of these equations. I shall have 
occasion also to refer to results, and further develop the theory contained in my 
memoir, “On the Transformation of Elliptic Functions,” Phil. Trans., t. CLXIV. (1874), 
pp. 397—456, [578], and the addition thereto, Phil. Trans., t. clxxxix. (1878), pp. 
419—424, [692]. 
I remark that, while I have only worked out the formulae for the cases n = 3 
and n = 5, and a few formulae for the case n = 7, the memoir is intended to be a 
contribution to the general theory of the pa/3-transformation; I hope to be able to 
complete the theory for the case n = 7. 
[* Ges. Werke, bd. i., pp. 295—318; in particular, p. 303.]