178 Dr. G. Child on the Production of [Apr. 27, 



quadrilateral. Then the nature of the change of coordination, by which a 

 new index figure is coordinated with the same stigma figure, is explained, 

 and bilinear and directional coordination introduced and illustrated by 

 applying them to deduce the usual formulae for the transformation of 

 Cartesian coordinates from oblique to oblique, and from oblique to polar. 

 This is followed by the most general theory of transversals cutting or inter- 

 secting upon any stigmatic curve, and by trilinear coordination. The 

 equation to a stigmatic point is then discussed, giving rise to classes of 

 stigmatic curves with bipunctual and tripunctual coordination. The 

 investigations on trilinear and tripunctual coordination contain generaliza- 

 tions of Professor Pliicker's ' Point and Line Coordinates,' by which their 

 precise geometrical meaning, even when "imaginary," and even in more 

 general cases than those " imaginaries " which he contemplated, becomes 

 manifest from the very form of the equations. 



Although details have been avoided as much as possible in the latter part 

 of the memoir, the writer hopes that sufficient has been given to enable 

 any mathematician to apply the theory with ease and safety to the gene- 

 ralization and linear realization of every theory on plane geometry which 

 has hitherto been propounded. The conception is equally applicable to 

 solid geometry, but will there require the algebra of quaternions, which, 

 being non-commutative, establishes a well-marked line of separation between 

 plane and solid stigmatics.. The writer has not found a trace of this gene- 

 ralization in the works of any previous author, but the relations, when 

 pointed out, appear too obvious to have escaped all notice hitherto. The 

 writer believes that in any case no complete theory, such as that presented 

 in this memoir, has been previously founded upon any similar conception. 



The Society then adjourned, over the Easter Recess, to Thursday, 

 April 27. 



April 27, 1865. 

 Major-General SABINE, President, in the Chair. 



Pursuant to notice given at the last Meeting, Sir Henry Holland pro- 

 posed, and Dr. Bence Jones seconded, His Royal Highness the Count of 

 Paris for election and immediate ballot. 



The ballot having been taken, His Royal Highness the Count of Paris 

 was declared duly elected. 



The following communications were read : 



I. " Further Experiments on the Production of Organisms in Closed 

 Vessels." By GEORGE CHILD, M.D. Communicated by Pro- 

 fessor PHILLIPS. Received March 30, 1865. 



The researches, an account of which is contained in the following paper, 

 are in continuation of those which, through the kindness of Prof. Phillips, 



