ON ELLIPTIC AND HYPERELUPTIC FUNCTIONS. 307 



preserved carapace of a small crustacean under the name of Mitliracites vec- 

 tensis, from the Greensand, Atherfield, Isle of Wight. I lately obtained six 

 specimens from the same locality, which upon comparison I found to agree 

 (so far as the figures and desciiption enabled me to determine) with Gould's 

 Mithracites ; but when I compared the specimens with the receut Mithrax, I 

 failed to discover the analogy, although the specimens since obtained appear 

 to oifer a decided affinity with the genus Hi/as. The discovery of these ad- 

 ditional examples will necessitate the reconsideration and redescription of the 

 genus Mithracites. 



Fortunately the abdomen and limbs of both male and female examples are 

 preserved; and the margins of the carapace are also well seen, 



Erom the Greensand, Isle of Wight, I have also obtained a new species of 

 Hemioon? (Bell), but larger than H. Cunningtoni. From the Hard Chalk, 

 Dover, I have received a new form of Enoplodytia, which I propose to call 

 E. scabrosa. 



Only one new species of Trilobite has to be noticed ; it was found at 

 Utah, and sent over by Mr. Henry S. Poole, Inspector of Mines, Nova Scotia. 

 I have referred it to the genus Olenus, under the name of Olenits utahensis. 

 It shows evidence of a median axis, apparently corresponding with the so- 

 called straight alimentary canal, noticed by Earraude. The matrix is com- 

 posed of a hydrated silicate of magnesia. 



This completes the list of new forms examined and determined by me, 

 some of which are already engraved for publication. 



Report on Recent Progress in Elliptic and Hyperelliptic Functions. 

 % W. H. L. Russell, F.R.S. 



Paet II. On the System of Hyperelliptic Differential Equations adopted by 

 Jacohi, Gopel, and Eosenhain. 



In this part the solutions of the hyperelliptic differential equations of 

 the first order, as given by Gopel and Rosenhain, will form the main sub- 

 ject which I desire to bring before my readers. They will ever possess 

 great interest, although surpassed in generality by the later researches of 

 Weierstrass, and the geometrical methods of Riemanu. The researches of 

 Gcipel and Eosenhain were nearly contemporary ; as, however, those of 

 Rosenhain are somewhat more elaborated than those of Gcipel, I shall com- 

 mence with an account of them, as contained in the ' Memoires de I'lnstitut, 

 par Divers Savants,' torn. xi. p. 361. Rosenhain begins his investigations 

 by giving formulse for the multiplication of four functions d appertaining to 

 elliptic integrals, and uses these as a starting-point for the corresponding 

 formulae in hyperelliptic functions. He then expresses these new functions d 

 in terms of two new variables, and shows that from the equations thus ob- 

 tained we can deduce the hyperelliptic differential equations. 



Section 1. — We commence with Roseuhain's multiplication of four func- 

 tions t) in the case of elliptic integral. His notation is as follows (it will 

 be observed that he uses the same notation we have been already familiar 

 with in Schellbach, except that his exponentials involve real quantities) : — 



X 2 



