Remarks on the Theory of Vanishing Fractions. 211 



formula on page 106. Now in all these examples the factors 

 V c" — d b", h d — c U, are each zero, and therefore in finding 

 the value of x from the equations (4), (5), page 105, where 

 multiplications are directly pei'formed by these factors, the ob- 

 jectionable zero process actually occurs. Let any one follow 

 the steps of the general elimination, page 105, with one of 

 the particular examples, and the fallacy will immediately 

 present itself. It is the direct incorporation of these foreign 



zero factors that gives to the results the form — , which they 



would not otherwise assume; and they furnish Professor 

 Young with another example of the zero processes which he 

 has " never seen." 



The expulsion of foreign zero factors, which are usually in- 

 troduced in an investigation by direct multiplication, is the 

 object of my fourth proposition. If they were allowed to enter, 



as legitimate factors, in mathematical reasonings, the form — 



might very evidently represent any species of quantity whatever. 

 The remarks of Bourdon on this symbol do not refer to what 

 it strictly represents, but only to its general indication, consi- 

 dering all cases without any regard to the legitimacy of the 

 process. What Professor Young states respecting the foreign 

 factors is altogether foreign to the subject in dispute, as the 

 general principles maintained in my essay expressly object to 

 the introduction of all such factors. The real statement of the 

 question is this : Is the value of a vanishing fraction indetermi- 

 nate or not when the zero factors, as in the ellipse question, 

 are not directly introduced, but arise spontaneously in the 

 investigation? and I have thoroughly discussed it in my former 

 papers. 



Professor Young's remarks concerning multiple solutions, 

 or what ought to have been properly termed indeterminate 

 solutions, is a mere quibble about words. I must frankly 

 confess that his important distinction between the two state- 

 ments is one in which I cannot perceive the slightest difference 

 in substance. It would be an idle waste of words to say any- 

 thing more about it. His observations on singular solutions, 

 however, call for some remark. No objection whatever has 

 been offered to the statement which he defends so cavalierly 

 in his present letter. It must be highly amusing to the readers 

 to perceive him calling on the aid of Lagrange in support of 

 a statement that no one, for aught I know, ever thought of 

 disputing. The true state of the case is as follows. Speaking 

 of the result of a general investigation furnishing every solution 

 to a problem, in his last letter he alluded to the well-known 

 fact of singular solutions not being comprised in the general 

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