102 REPORT—lSrO. 



" 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 fii-eball at 

 gh gym p_-j[._ observed at the Isle of Skye, and a more than ordinarily bright 

 meteor at 11"* 16™ 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 -q Persei, at East Tisted (Mr. F. Howlett) near 

 er 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 ^ Persei, and at 

 Hawkhurst between tj Persei and e Cassiopeiae. At Manchester the radiant 

 region on the nights of the 6th-9th of August appeared to Mr. Greg to 

 occupy an elongated space between I- Persei and e Cassiopeiae. 



Report 011 Recent Progress in Elliptic and HypereUiptic Functions. 

 By W.B.. L. Russell, F.R.S. 



Part III. 



Section 1. — In this division I propose to consider modular equations, and 

 some subjects connected with elHptic functions, omitted in the Second Part. 

 The higher portions of the theory of modular cqiiations, which are inti- 

 mately 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. E. 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, Hcrmite, and Joubert, which 

 are closely connected with these investigations of Sohnke. 



I shall employ in the following pages fi and v instead of u and v, as used 

 in the ' Fundamenta Nova,' to prevent (m) occurring in two different senses 

 in the same investigation. 



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

 order : — 



d)f v — p.^ dx 



Vl^y-VI^^7p~Kl-w>'')* Vl— a-Vl-uV 

 if 



y v^(l-^.'-^) + ;uvV + »'0 ("-yu'K + A''»''(''-A*'>'' 



waere 



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



