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equation is multiplied or divided by a quantity of the order n, the 

 index of equality must be increased or diminished by n. Various 

 cases are given in which such results as now present anomalies 

 are reduced under formal law, and others which would be absolutely 

 rejected are shown to be capable of consistent interpretation. 



The formal law of connexion of the different states, of which fini- 

 tude (with the index 0) is only one, is that the order m stands to 

 finite quantity in all respects as finite quantity to oo OT . Hence, so far 

 as 1 and 1 + are simultaneous as well as equal, so far oo and co + a 

 are simultaneous as well as equal. And if 0(1) = O $(1 +0) be a uni- 

 versal law, so must be 0(co ) = 0(co + a). Further, oo — oo must 

 be, formally speaking, wholly indeterminate, even when it is a case 

 of x — x. 



In relation to such indeterminate forms as go — co , -g-, &c, Mr. De 

 Morgan contends that their formal and d, priori character is that of 

 indeterminateness ; and that the choice between determinate and in- 

 determinate character, which so often occurs, is dictated by the matter 

 of the problem, the determinate value being dictated by the laws of 

 algebra. The index of equality, for instance, may be the means of 

 decision : an example is given in which one equation belongs to two 

 different problems, but with different indices of equality ; in one £ 

 is determinate, in the other wholly indeterminate. 



In assigning co or as values, it is often necessary to assign rela- 

 tions of order. When a quantity passes from positive to negative, 

 or the converse, through or oo , it passes through every order of 

 or co ; and this even when the passage is from one phase of or co 

 to another, of different signs. Thus, the orders being powers, x 

 cannot pass from — a.0 m to +a.0 m , without passing through 

 even 0°° . 



Mr. De Morgan insists upon one of two things : either, the aban- 

 donment of the separate use of and oo, except only in the retention 

 of the former symbol to represent A — A ; or, the introduction of dif- 

 ferent orders, and the free use of the comparisons of those orders. 

 For himself, he prefers and adopts the latter alternative. 



The principle of limits is considered as a formal law of algebra, 

 but not to the exclusion of every other result. If a constant, for 

 instance, have the value A up to x=a exclusive, it has A for one 

 value when x=a. If the constant be transitive, that is, if it be 

 always =B after x=a, then x=a gives both A and B for the con- 

 stant, and, as a fact hitherto observed, its value from calculation is 

 ^(A + B). This observed fact Mr. De Morgan believes he connects 

 with the principle of limits, making it a necessary consequence of 

 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 co , it is shown that it is cor- 

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

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

 epoch of transitiveness the value of <j>(a + 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 



