29 



From this last result the existence of n roots and no more for an 

 equation of n dimensions is the immediate result. 



Several well-known theorems are deduced from this view of the 

 subject, and are given as illustrations. 



The actual curves are traced, corresponding to the various cases 

 of the quadratic, the cubic, and the biquadratic equations, and to 

 the equation x n — 1=0. 



In the conclusion of the memoir it is remarked that the -esults 

 obtained are not exclusively applicable to the case of algebraic 

 equations, and the methods are applied to the case of f(x)= sin x. 



The author trusts that the contents of this memoir, though not 

 adding to the number of known theorems, may yet be useful as 

 putting the subject in a new light, and as furnishing a method of 

 demonstrating the existence of the roots of algebraic equations more 

 simple and direct than any other which he has seen. 



Cases of Morbid Rhythmical Movements, with observations. By 

 G. E. Paget, M.D., Fellow of Caius College and of the Royal College 

 of Physicians, London. 



Seven cases were related. The movements were vibratory, rota- 

 tory, bowing, &c. In some of the cases they were incessant ; in 

 others paroxysmal ; and in others again they were of both kinds, 

 the predominant movement being replaced at intervals by distinct 

 paroxysms. 



On a comparison of these cases with the few others on record, 

 and with the experiments of Flourens and Majendie, it was inferred 

 as probable, that one class of the movements, viz. the rotatory, de- 

 pended on disorder in the cerebellum or its transverse commissure, 

 the pons. With regard to the other movements, it appeared that 

 there were no sufficient grounds for even a probable conjecture as 

 to the particular part of the encephalon, the excitement or disorder 

 of which might act as an immediate cause of the movements. 



The remote causes were such as under other circumstances are 

 known to excite the common convulsive diseases, such as chorea anp 

 epilepsy. These remote causes were in most cases eccentric. 



November 9, 1846. 



On the Structure of the Syllogism, and on the application of the 

 Theory of Probabilities to Questions of Argument and Authority. By 

 Professor De Morgan. 



The object of this paper is twofold : first, to establish two distinct 

 theories of the syllogism, both differing materially from that of Ari- 

 stotle, and each furnishing a general canon for the detection of all its 

 legitimate forms of inference ; secondly, to investigate the mode in 

 which the distinctive character of the two great sources of convic- 

 tion, argument and authority, affects the application of the notion of 

 probability to questions not admitting of absolute demonstration. 



