534 EEPOKT — 1881. 



6. Some neiu Theorems on Curves of double Civrvature. 

 By Professor Sturm. — See Reports, p. 440. 



6. On Congruencies of the Second Order and Second Class. 

 By Dr. T. Archer Hirst, F.E.S. 



7. Sur les faisceaux de forme hiquadratique binaire ayant une meme 

 Jacobienne. Par Cyparissos Stephanos. 



8. On a Diagram connected with the Transformation of Elliptic Functions. 

 By Professor Cayley, F.E.S. 

 The diagram relates to a known tlieorem, and is constructed as follows. Con- 

 sider the infinite half-plane y=+ \ draw in it, centre the origin and radius unity, 

 a semicircle ; and draw the infinite half-lines x = -^, and x = ^; then we have a 

 region included between the lines, hut exterior to the semicircle. The region in 

 question may he regarded as a cm-\-ilinear triangle, with the angles 60^, 60°, and 0°. 

 The region may be moved parallel to itself in the direction of the axis of x, tkrough 

 the distance 1 ; say this is a ' displacement ' ; or we may take the ' image ' of the 

 region in regard to the semicircle. Performing any number of times, and in any 

 order, these two operations of making the displacement and of taking the image, 

 we obtain a new region, which is always a curvilinear triangle (bounded by circular 

 arcs) and having the angles 60°, 60", 6° ; and the theorem is that the whole series 

 of the new regions thus obtained completely covers, without interstices or over- 

 lapping, the infinite half-plane. The number of regions is infinite, and the size of 

 the successive regions diminishes very rapidly. The diagram was a coloured one, 

 exhibiting the regions obtained by a few of the successive operations. 



The analytical theorem is that the whole series of transformations, to into 



<L?L+J^ ^l^ere a, /3, y, 8 are integers such that aS-/3y = l, can be obtained by 

 •y o) + d 



combination of the transformations u> into o> + 1 and « into — . 



9. A 'partial Differential Equation connected with the simplest case of 

 AbeVs Theorem. By Professor Cayley, F.B.S. 



Consider a given cubic curve cut by a line in the points (.ij, y^), (.i\, i/^), 

 (a.-,, 3/3) ; taking the first and second points at pleasure, these determine uniquely 

 the third point. Analytically , the equation of the curve determines y^ as a 

 function of .r„ and y^ ^■s a function'of x^ : writing in the equation x^ = Xx^ + (1 - X).r2 

 y =Xm +(1 — X) y.„ we have X by a simple equation, and thence x^; viz. x^ is 

 found as a function of .r,, .r,,, and of the nine constants of the equation. Hence 

 forming the derived equations (in regard to .I'l, .r,) of the first, second, and third 

 orders, we have (1+2 + 3 + 4 =)10 equations from which to eliminate the_ 9 

 constants ; x^ considered as a function of x^, x„ thus satisfies^ a partial differential 

 equation of the third order, independent of the particular cubic curve. 



To obtain this equation it is only necessary to observe that we have, by Abel's 

 theorem : 



Xj X2 X3 

 ■where X^ is a given function of .rj and y,, that is of x^ ; and X^, X^j^xe. the like 

 functions of x^ and x^ respectively. Hence, considering x^ as a function of x^, x^, 



■we have 





dx^ Xj dx^ X,_ 



•2 



