1878.] 



Prof. J. Casey on the Equations of Circles. 



417 



posterior part of the fourth, ventricle, passes forward, and at the point 

 above-named ascends on the external edge of the medulla, and forms the 

 commissure of the tecta lobi optici, at the base of their longitudinal 

 ridges above described : this commissure also forms a decussation of 

 fibres beneath the ganglion of the trochlearis. 



The anterior commissure of the brain is situated between the 

 cerebral lobes. 



The posterior commissure is situated on the anterior part of the 

 floor of the ventricle of the optic lobe behind the third ventricle. 



A deeper commissure connects the region on each side of the third 

 ventricle passing through the hypoaria in front of the inf undibulum . 



Two transverse commissures exist in the spinal cord, one ventral, 

 one dorsal. 



The ventral longitudinal columns of the cord contain two fibres of 

 gigantic size, one on each side ; these decussate opposite the origin 

 of the trifacial on the floor of anterior part of the fourth ventricle ; 

 the longitudinal columns pass to the internal part of the floor of ven- 

 tricle of the optic lobe, about the posterior end of the fourth ventricle. 



The lateral columns of the cord pass forward, and are lost outside 

 the last. 



IV. " On the Equations of Circles." (Second Memoir.) By John 

 Casey, LL.D., F.R.S., M.R.I.A., Professor of Mathematics 

 in the Catholic University of Ireland. Received May 10, 

 1878. 



(Abstract.) 



In the year 1866 was published in the " Proceedings of the Royal 

 Irish Academy" a paper " On the Equations of Circles," which con- 

 tained extension of many known theorems. Thus it was proved in it 

 that the same forms of equation which are true for a circle inscribed 

 in a plane or spherical triangle hold also when the right lines in the 

 one case, or the great circles in the other, are replaced by any three 

 circles in the plane or sphere, and it was shown that the transformed 

 equations represented the pairs of circles which touch the three 

 given circles. The results for circles on the sphere were still further 

 extended, namely, to conies having double contact with a given conic. 

 The paper contained, in addition to these fundamental investigations, 

 many collateral ones on allied subjects. 



The memoir, of which I now give an abstract, extends the results of 

 the foregoing paper to a polygon of any number of sides inscribed or 

 circumscribed to a given circle. It is proved for the case of circum- 

 scribed figures, that the sides of the polygon may be replaced both on 

 the plane and sphere, by circles touching the given circle; and 



