"There was a marked paucity of meteors on and about the 10th; and the August meteoric shower appears to be approaching its minimum, which, as observed in these Reports for 1867 and 1869, might be expected [dating an eight-year period, observed in previous minima, from the minimum of 1862] to take place about the present year. On the present occasion the greatest number occurred on and after midnight of the 6th, preceded by a fireball at 9h 57m P.M. observed at the Isle of Skye, and a more than ordinarily bright meteor at 11h 16m P.M., at Birmingham. There was no change of position of the radiant-points on successive nights, but a continuation of the Perseids, and other centres, with a simple variation in activity.”

The radiant-point on the night of the 10th appeared at London (Mr. T. Crumplen) to be near 7 Persei, at East Tisted (Mr. F. Howlett) near o Persei, and at Hawkhurst (Mr. A. S. Herschel) near a, y Persei; on the night of the 11th it appeared to be, at London nearer to x Persei, and at Hawkhurst between n Persei and e Cassiopeia. At Manchester the radiant region on the nights of the 6th-9th of August appeared to Mr. Greg to occupy an elongated space between k Persei and e Cassiopeia.

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Report on Recent Progress in Elliptic and Hyperelliptic Functions. By W. H. L. RUSSELL, F.R.S.

PART III.

Section 1.-In this division I propose to consider modular equations, and some subjects connected with elliptic functions, omitted in the Second Part. The higher portions of the theory of modular equations, which are intimately connected with the theory of numbers, have been already treated by Professor Smith in his valuable report on that branch of mathematics. On the other hand, Professor Sohnke's important paper on modular equations was very slightly noticed by Mr. R. L. Ellis, and therefore, although much earlier in date than the other papers which form the subject of this Report, may well be considered here as an assistance to the reader who is disposed to enter on the researches of Messrs. Kronecker, Hermite, and Joubert, which are closely connected with these investigations of Sohnke.

ν

I shall employ in the following pages μ and instead of u and v, as used in the Fundamenta Nova,' to prevent (u) occurring in two different senses in the same investigation.

Jacobi has given the following theorem for the transformation of the 5th order :

This last equation is called the modular equation of the 5th order.

4

4

Putting =√, μ=√k, the general problem of modular equations, which we have to solve, is the following :—

To determine the relation between λ and k, or μ and

so that

√ 1—y2 √1—λ2y2 ̄ ̄M√1−x2 √1−k2x’

M being a constant multiplier, and y and a connected by the relation

We easily derive the following theorem from the Fundamenta Nova :'— 1-k sin am nu=

and substituting in this the values of the factors in the numerator derived from the equation,

transforming by the formula of page 37 of the Fundamenta Nova,' and determining the constant multiplier by putting n=o in both sides of the equation, we have

where, however, when m=0, IIm' must be substituted for Пm'.

From this Sohnke deduces that symmetrical functions of the quantities 4mK+4m'iK'

sin coam

n

when m and m' have the values just assigned, are

rational and entire functions of k.

Section 2.-We know that

v="{sin coam 4w sin coam 8w

sin coam 2(n-1)w};

and it appears, from the Fundamenta Nova,' that if we put in this equation successively the (n+1) values,

we shall have all the possible values of this expression. The values of may therefore be represented by the following expression :

where m' signifies one of the quantities 0, 1, 2, 3.

(n-1), and m is

unity, except when m'=1, when it is both unity and zero.
We immediately deduce from the Fundamenta Nova' the equation

From this Sohnke proves that the (n+1) values of (») may be derived from

by substituting for q successively in this equation the (n+1) quantities

a being any of the n roots of unity.

The proof, although long, presents no particular difficulty, and depends on transforming the factors in the continued products by means of the theorem that 2nr+4m'p (when n is a prime number, m' one of the numbers 1, 2, 3 n-1, r all numbers from zero to infinity, p all numbers from 0 to

N

2

lected.

is an expression representing all even numbers, the sign being neg

Section 3.-It appears from this investigation:

A. That the modular equation is of the (n+1)th degree.

B. That the coefficients of the equation, when arranged in powers of are rational and entire functions of με

C. That the last term of the modular equation is of the form +μ2+1 if (n) is of the form 8r+1, +1 if n is of the form 8r+3.

This is deduced by Sohnke from the observation that it is a consequence of the multiplication of elliptic functions that all the roots should have the same sign as the quantity

D. The modular equation is unchanged when k and A are interchanged, therefore the highest power of (μ) cannot exceed (n+1).

E. We have already seen that μ is of the form /2f(q). One value of () must therefore be of the form 2/qf(q"). If we substitute this in the modular equation, the irrationality must disappear. Hence in any term of the modular equation aμm,, we must have/qm.qrn=q/qt, where t is constant for every term. Hence m+rn=88+t, and therefore the modular equation must be made up of terms of the form

8

F. Since +1 is of the form q+1R(q), and μ of the form +1R2(q), we see the irrationality to be the same in each case. Hence, as the equation necessarily admits of a term cμ"+1, it must also admit of a term of the form όμν.

and

G. Since the modular equation remains unaltered when k and 7 are interchanged, it follows that it must also remain unaltered when μ interchanged. The modular equation is of the form

2n+1+...+aμv+μ2+1=0

vare

",

if n is of the form 8r+1. Here we must manifestly interchange μ and r, as the equation cannot be reproduced if μ is placed instead of ", and — instead of μ. On the other hand, the modular equation is of the form

μ

+αμν-μη+1=0

μ

if (n) is of the form 8r+3. Here we must place μ instead of v, - instead of μ, as the equation cannot be reproduced if we interchange μ and v. H. Hence Sohnke shows that the coefficients of μm, and μ are equal always in magnitude, although differing in sign when n=8r+3, and Р is Also that the coefficients of up and un+1-m,n+1−p are always equal in magnitude, although differing in sign, when n=8r±3.

even.

J. Lastly, Sohnke proves that when μ=1, the equation necessarily takes the form (+1)" (v-1)=0 when n is of the form 8r+3, and (v−1)n+1 when (n) is of the form 8r+1.

Section 4.-The method of ascertaining the form of the modular equation now becomes manifest.

We determine the indices of μ and by E. Then H, J give us relations between the coefficients, which greatly diminish their number considered as independent quantities. Finally, we determine the remaining coefficients by substituting the values of μ and expanded in terms of q in the equation, and then equating the coefficients of the powers of q thus obtained to zero. This method is fully illustrated by Sohnke by an example. He has also added a modification of the process, which will be found useful in practice. Section 5.-The discriminant of the modular equation is of the form

For a proof of this the reader is referred to the concluding section of Professor Betti's Monografia on Elliptic Functions, contained in the third and fourth volumes of the Annali di Matematica,' which I am now going to bring under the notice of the reader. Professor Betti has founded his theory on the geometrical basis adopted by Riemann and his followers, and which it is not my object to consider in the present Report. I shall therefore explain at once the connexion between the notation in the Monografia with that we have already employed, and so lead the way to some new aspects of elliptic functions.

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Putting w=2K, w'=2iK', and therefore q=e we have, according to Professor Betti's notation,

This notation allows us to make use of the following definition, which is of fundamental importance throughout Professor Betti's memoir :

when

0,,

(5)

πί

Section 6.-Having thus explained the notation, we come to the following theorem given by Professor Betti (A. D. M. 3. 123):

where

20μ, v (z+w)Ðμ', v'(z—w)Əμ—μ', o(0) 8o, v—v'(0)=
Po, o (2)+(-1) Po, 1 (≈) + P1,0 (≈)+(−1)" P1, 1(2),

Pn, e(2)=0μ+n, v+e (2) Ɖμ'+n, v'+e(2) Əμ−μ'+n, e(W) On, v-v'+e(w).

(1) ・・

The roots of the entire functions Oμ, v (z+w) Əμ', v' (z-w) are respectively

of the form

and the theorem is shown to depend on the proposition that these are also roots of the expression

F(x)=Po,o(z)+(−1)" Po, 1 (2) + P11(≈)+(−1)“ P11(2),

(2)

The reader will find no difficulty in proving this by means of the formulæ of last section, and the expressions for the periods given by Schellbach, p. 34.