72 Cambridge Philosophical Society. 



the universal truth of that principle ; and hence he holds that it may 

 be stated as a theorem, under the name of the principle of mean 

 values. Various uses of this principle are given. Further, in assu- 

 ming the free use of the orders of and oo , it is shown that it is cor- 

 rect to say that the constant passes from A to B while h, x being 

 a + A, passes through the phases of 0. So that, for instance, at an 

 epoch of transitiveness the value of $(a -f 0) is dependent upon the 

 form of 0. The brevity of an abstract prevents the statement of 

 those cautions under which such use of language is introduced. One 

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

 transitive (or, as commonly said, discontinuous) at x=a, the equa- 

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

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

 differential coefficient infinite. 



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

 their observed values, sinoo = and cosoo=0, both from the prin- 

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

 0(oo -fa) = (/»oo. From the same principles follows the equation 

 ( — 1)°° =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 oo — 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 

 be 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, d, 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, d, 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 



