MATHEMATICS: B. I. MILLER 
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formations of / and r, which leave the form a" unaltered, i.e., the sixty 
transformations of the icosahedral group applied simultaneously to / 
and r, the parameter of the doubly-covered conic R2 and the modulus 
of the elliptic quintic curve E^, leave Ui unaltered. 
Consider now Bianchi's integral. It is defined as 
where C is a constant, u and v any two expressions linear in and the 
denominator is the functional determinant of <po, <pi, ^, u and v. For 
a particular choice of u and v the integral assumes the simple form 
where the x's are subject to the relations (pi. Different expressions for 
U can be obtained by making different choices for u and v. Hence 
there is no unique form for U as there is for Ui. The integral U assumes 
various conjugate forms under the Group G50 of collineations on the x^s, 
and also under the transformations of a. 
So the integral Ui seems to have an advantage over U in its simplicity 
of form, its uniqueness, and its invariancy under transformations. 
By a study of the integral Ui itself some interesting results are de- 
rived. The modular equation connecting r and /, the absolute invariant 
of Uiy can be deduced as the result of the binary syzygy of lowest weight 
connecting the concomitants of a^. The requirement that the Riemann 
surface attached to the modular equation be regular leads to the modu- 
lar equations associated with the regular bodies. It is then possible to 
eliminate the more tedious individual proofs used by Bianchi in the 
discussion of the moduH of £3 and E(, to show that these moduH are the 
tetrahedral and icosahedral irrationaHties respectively. In fact the al- 
gebraic discussion carried out once for is complete for factor groups of 
genus zero, which have been discussed by Klein,^ i.e., those isomorphic 
with the groups associated with the regular bodies, namely, the one 
dihedral group Ge and the tetrahedral, octahedral, and icosahedral 
groups. 
1 Klein-Fricke, Vorlesungen uber die Theorie der EllipHschen Modulfunctionen, Bd. 2, 
Abschnitt 5. 
2 Bianchi, tlber die Normalforaien dritter und fiinfter Stufe des elliptischen Integrals 
erster Gattung, Math. Ann., Leipzig, 17, 234-262, (1880). 
3 Klein-Fricke, loc. cit., vol. 1, pp. 339 ff. 
