1898.] differential equations defining periodic functions. 521 



where P = 2« 3 + |X 6 « 2 + 2xy + \\ b x ~2z + 2^ + £ X 4 , 



Q = 2x 2 y + \\ 6 xy + 2xz — x% + y~ + j\ 5 y + v + i^s; 



R = 2x°z + %\xz + yz-xr) + l \ 5 z - £, 



S = 2xz° + %\z 2 - 2z<n + | X , 



T = 2xyz + \\*yz + 2z 2 - yv - z\- + {\ , 



U= 2xf+^\ ti y + 4yz - 2yZ+2£+ i\. 



Each of the three relations is, when reduced, of the fourth 

 degree in the aggregate. We find other relations, also of the 

 fourth degree, which are deducible from the three above. Of 

 these two have forms which deserve to be recorded : — 



y + i\ 6 , 2x + ±\ = 0, 



2%-2z + ±\ 4 , y + i\ 



t V 



-x, 1, =0; 



2£-2z + ±\ 4 , y + l\ 



& v 



if in the first of these determinants we omit x, y, z and replace 

 |, 7], £ by x, y, z, it becomes identical with the Kummer nodal 

 quartic obtained in § 3 ; an independent set of three of these 

 equations is indeed to be regarded as the generalisation of the 

 Gopel relation to the hyperelliptic case p = 3. For any value of 

 p the generalisation should consist similarly of a set of \p>{p — 1) 

 equations in \p{p + 1) Abelian functions. There is a generalisa- 

 tion in another direction in which there enter 2^ — 1 Abelian 

 functions — holding however for the general case of p variables. 



(3) On the Total Eclipse of the Sun 1898 Jan. 22. By 

 H. F. Newall, M.A., Trinity College. 



A general account was given of the observations made during 

 the recent eclipse, and photographs were exhibited, shewing (i) the 

 general appearance of the corona, (ii) the spectrum of the sun's 

 limb as photographed with prismatic cameras by Sir Norman 

 Lockyer's party, and by Mr Evershed, (iii) the spectrum of the 

 sun's limb as photographed with a slit-spectroscope by Mr Newall, 

 and a diagram was shewn illustrating the distribution of glowing 

 ' coronium ' as observed by Mr Newall with a slitless grating 

 spectroscope. 



Capt. E. H. Hills, R.E., exhibited and described the photo- 



