126 



result, however, may be brought forward. When the function/* is 

 transitive (or, as commonly said, discontinuous) at #=«, the equa- 

 tion <p(fa) = (<pf)a no longer necessarily exists. But this, as is 

 pointed out, is what may happen at any value of oc which makes a 

 differential coefficient infinite. 



On the question of sin oo and cos oo , Mr. De Morgan deduces 

 their observed values, sin go = and cos 00 = 0, both from the prin- 

 ciple of mean values, and from the formal truth of the equation 

 0(co -f-«) = </>co . From the same principles follows the equation 

 ( — I) 00 =0. In this case, however, and in all which come under 

 the principle of mean values, the absolute necessity of the results is 

 not affirmed. They are the alternatives of indeterminateness. But 

 in thus representing them, Mr. De Morgan does not concede more 

 than he conceives must be conceded with respect to 00 — oo , £, and 

 the like. 



On the question of series, Mr. De Morgan contends that all the 

 uncertainty and danger of divergent series belongs equally to con- 

 vergent series, in every case in which the envelopment is unknown. 

 On this part of the subject he adds to the arguments of a former 

 paper, and insists upon the superior safety of the alternating series, 

 in which the terms are alternately positive and negative. 



Without going further into details, the purport of this paper may 

 he stated as follows. Algebra, using the term in the widest sense, 

 ought to be, and is approaching towards, a science of investigation, 

 and a symbolic art of expression, of which the laws are strictly and 

 without exception incapable of failure, suspension, or modification. 

 The formal laws under which such a result is to be obtained, though 

 laid down in the first instance by extensive induction, of which many 

 steps are accompanied by difference of opinion, will at last be received 

 and admitted as parts of the definition of the science, ci priori. The 

 existing defect of the science is an imperfect formalization, arising 

 from the want of views of sufficient extent, and leading to material 

 distinctions, that is, to exceptions dictated, a posteriori, by the re- 

 sults of particular cases. Such exceptions have in many instances 

 been brought within rule by further consideration ; and it is con- 

 ceived that the same thing will happen at last in all cases. The 

 paper is an attempt to examine the principal outstanding difficulties 

 (those connected with the definition of integration excepted) with 

 reference to the question how far they may arise from imperfect 

 conception of formal laws. That there is to be a formal science, is 

 positively assumed, and made the basis of the attempt : how far any 

 suggestion contained in the paper is a valid step towards it, is treated 

 with doubt and left to opinion. 



May 9, 1853. 



Mr. Hopkins, F.R.S. &c, the President of the Society, gave an 

 account of some experiments for the determination of the tempera- 

 ture of fusion of different substances under great pressure ; and on 

 the application of the results to ascertain the state of the interior 

 of the earth. 



