1893.] 



On the Ichtliyosauria and Sauropterygia. 149 



lines lying on a cubic surface. This theorem may be deduced from 

 the equation 



xyzu = (x + aT) (y + bT) (* + cT) (w 



where T = ax + fiy + ^z + Bu-, and a, Z>, c, d, a. /?, 7, 8 are constants. 

 The equations of twelve lines on the surface are evident. 



This paper shows how the remaining fifteen straight lines on the 

 surface may be obtained by means of nothing higher than quadratic 

 equations, and determines which of these lines intersect each other. 



The paper then proceeds to give a graphical method of represent- 

 ing all the intersections of the twenty-seven lines on a cubic surface 

 by means of a plane diagram, which admits of many interesting 

 transformations. 



By the help of such diagrams some of the known relations of the 

 twenty-seven lines to each other are deduced, and some theorems 

 with respect to the lines, which it is believed are new, are established ; 

 for instance, the number of closed quadrilaterals, pentagons, and 

 hexagons on the surface is determined, as well as the number of 

 ways in which nine triple tangent planes can be drawn to pass 

 through all the twenty-seven lines, and the number of ways in which 

 twelve of the lines can be chosen, so that they are the intersection 

 of two tetrahedrons in perspective. 



XII. " Further Observations on the Shoulder Girdle and Clavi- 

 cular Arch in the Ichthyosauria and Sauropterygia." By 

 H. Gr. SEELEY, F.R.S. Received May 25, 1893. 



On January 18, 1892, I communicated to the Royal Society 

 observations on the nature of the shoulder girdle and clavicular arch, 

 in Sauropterygia, which were read on February 18, and published 

 in the Proceedings on June 25, 1892. These studies had grown out 

 of the examination of new remains of Anomodont R.eptiles, which I 

 obtained in South Africa ; and were the result of an endeavour to 

 gain a knowledge of structures in which the shoulder girdle in extinct 

 Reptilia admitted of detailed comparison with those materials. I had 

 made examination of the same region of the skeleton in Plesiosaurs 

 and Ichthyosaurs, and communicated the results to the Geological 

 Society, which were published in the Journal of that Society in 

 November and December, 1874. 



In the paper of 1892 I endeavoured to correct, enlarge, or justify 

 interpretations previously given. One aspect of this revision led to a 

 controversial paper, challenging some points of interpretation which 

 occur among the facts in my contribution. It is entitled " On the 

 Shoulder Girdle in Ichthyosauria and Sauropterygia," by J. W. Hulke, 



