145 



theory of elliptic integrals was developed by a method the inverse 

 of that pursued in establishing the formula of common trigonometry. 

 In the latter case, the geometrical type was given — the circle — to 

 determine the algebraical relations of its arcs. In the theory of 

 elliptic integrals, the relations of the arcs of unknown curves are 

 given, to determine the curves themselves ; this is the principal 

 object of the present communication. 



The problem resolves itself into twelve distinct cases, depending 

 on the magnitude of the parameter, and the sign with which it is 

 affected ; out of the discussion of these cases arise many new and 

 important relations of elliptic integrals. It would excite little in- 

 terest to give the bare enunciations of those theorems, and a mere 

 outline of the methods by which they are established would be un- 

 intelligible. Xot the least interesting of those theorems is the pro- 

 position, that it is always possible to express an elliptic integral of 

 the first order as the sum of two elhptic integrals of the third order, 

 with parameters which are conjugate, reciprocal and imaginary. 



The author hopes, in a future communication to the Royal So- 

 ciety, — the present having grown under his hands beyond the limits 

 he anticipated — among other points, to extend his researches to the 

 case of elliptic integrals with imaginary parameters, and to show 

 the true geometrical meaning of such expressions. It will also be 

 shown, that imaginary expressions may be found for a logarithmic 

 elliptic arc analogous to the well-known imaginary exponential ex- 

 pressions for the sines and cosines of circular arcs. 



A paper was in part read, entitled, "Further Researches into the 

 Structure, Development and Functions of the Liver." By C. Hand- 

 field Jones, M.D., F.R.S. Received November 19, 1S51. 



January 29, 1S52. 

 COLONEL SABINE, R.A., TreastTrer, in the Chair. 



The reading of Dr. Handfield Jones' paper, " On the Structure 

 of the Liver," was resumed and concluded. 



Dr. Leidy and Profess-'^r Retzius, with ]Mulier, "SVeber and 

 Khronenberg, m-^intain the existence of plexuses of ducts in the 

 parenchyma of the liver containing the cells in their tubes. Some 

 other anatomists, especially Gerlach, believe the ducts to be pro- 

 longed into the lobules of the parenchyma, under the form of mere 

 intercellular passages without walls. 



Injections of acetate of lead in saturated solution, throWn into the 

 ductus communis choledochus, produce appearances which seem to 

 confirm the latter view. The author, however, believes them to be 

 fallacious, and that the ducts really terminate, as he has described them 

 in his former paper, by closed extremities, either rounded and even, or 



