400 



SCIENCE. 



[N. S. Vol. XIV. No. 350. 



which are satisfactory, and the heliocentric 

 distances and the arguments of the latitude 

 at the epochs of the three observations are 

 computed as in the method of Gauss. 



Let u^, u^, u^, r^, r^, r^ represent the argu- 

 ments of the latitude and the heliocentric 

 distances at the epochs of the three obser- 

 vations. Then the parameter, p, is defined 

 by the equation 



where r^ is expressed as a series in u whose 

 radius of convergence is determined as a 

 function of u^ and e. It is shown how the 

 coefficients are to be found. The eccen- 

 tricity, e, and the longitude of the perihe- 

 lion from the node, w, are given by 



f p — T" 

 6 sin (^t^ — uj) = \ -^-- — ^ cos (u^ — u^) 



L ^1 



■^ -^ 1- cosec (u„ — if,), 



6 COS (itj — w) = 



p — r^ 



The time of perihelion passage is deter- 

 mined from the law of areas. 



' On the Modular Functions associated 

 with the Riemann Surface 



s^ = 2(s — l)(z — x)(z — y) ' : 



Dr. J. I. Hutchinson, Cornell University. 

 The object of the present paper is to ex- 

 tend the results obtained by Picard in his 

 memoir ' Sur des fonctions de deux vari- 

 ables independantes analogues aux fonc- 

 tions modulaires ' (Acta Math., II., p. 114). 

 Picard considers in the first place the 

 integrals of the first kind, and in particular 

 the moduli of periodicity of the normal 

 integrals. By changing the values of x, y 

 in a continuous manner so as to return 

 finally to their initial values, the moduli 

 undergo a linear transformation, which 

 can be represented by a linear transforma- 

 tion on two parameters, u, v, in terms of 



which all the moduli are rationally expres- 

 sible. 



These transformations forming an infinite 

 group G can be generated by five special 

 ones S^ S^ ... S^, the explicit equations for 

 which were given by Picard in' a subse- 

 quent paper (Acta Math., Y.). 



The two variables x, y are then auto- 

 morphic functions of u, v, and all functions 

 belonging to the group can be rationally 

 expressed in terms of these. 



According to theorems previously ob- 

 tained by Picard, there exist functions 

 possessing a pseudo-autoraorphic character, 

 exactly analogous to the fuchsian theta 

 functions which Poincare uses in con- 

 nection with the automorphic functions of 

 a single variable. 



These functions can be constructed out 

 of the theta constants. In order to do this 

 it is necessary to determine the effect of the 

 transformations of the group G on the 

 latter, which is accomplished by means of 

 the transformation theory of the theta 

 functions. A table is constructed by means 

 of which pseudo-automorphic functions can 

 readily be constructed. 



' Some Future Solar Eclipses, in particu- 

 lar that of June 8, 1918, total at Denver ' : 

 Professor F. H. Loud and Mr. L. E. Inger- 

 soLL, Colorado College. 



The tables used in computing the circum- 

 stances of the eclipses herein discussed are 

 those ' On the Recurrence of Solar Eclipses ' 

 published by Professor Simon ISTewcomb in 

 1879. After some remarks upon the limits 

 within which the errors of such a compu- 

 tation may be expected to fall, the results 

 of w^hat seems the preferable combination 

 of Professor Newcomb's tabulated data are 

 stated as follows : 



On June 8,1918, the moon's shadow passes 

 across the United States from northwest to 

 southeast, covering Denver from 4*" 22"" 59* 

 P.M. to 4'^ 24°' 23'— a period of 1"" 24= ; while 

 on the central line the duration is 1"" 33'. 



