in rcplj/ to Professor Young. 21 



be assured in such cases that we obtain a complete solu- 

 tion as we necessarily do in perfect processes. The process 

 in the ellipse question is founded on general reasoning as 

 far as the vanishing fractions, and those fractions ought there- 

 fore to supply all the values that can satisfy the original equa- 

 tions. I still hold myself justified, therefore, in maintaining 

 that Prof. Young not only involves himself in a " palpable 

 inconsistency," but that he also takes an imperfect view of the 

 " theory of analytical results," when he denies the competency 

 of the results of the ellipse question to furnish the requisite 

 values, and at the same time agrees to receive them from the 

 original analytical conditions. 



With respect to the quadratic equation, advanced at page 

 519, the common process of resolution is perfectly general 

 when it is considered that the radical surd is capable of sus- 

 taining either the positive or the negative sign. The roots 

 4, f , presented by the involved quadratic, rigidly satisfy the 

 original aiialytical condition ; for the radical term may, analy- 

 tically, be interpreted either as being + or — , and so far as 

 the analysis is concerned it is perfectly immaterial which in- 

 terpretation is adopted. Thus the result 4 satisfies the con- 

 dition 2:r+ \/ x^ — 1 = 5 with the same analytical strictness 



that it satisfies ^ x— \/ x^—1 = 5 ; the only difference is that 

 in the one case the + and in the other the — interpretation 

 of the root is not complied with. It may be observed, how- 

 ever, that the nature of a problem originating such an equa- 

 tion may exclude all — interpretations, and therefore limit the 

 calculation to the + value of the root. In such a case the po- 

 sitive information, obtained from the analytical result, would 

 be that the numbers 4, f furnish every solution to the original 

 analytical condition ; the rejective information, suggested by 

 the nature of the problem, would be the exclusion of both 

 roots as not satisfying the implied condition of + interpreta- 

 tion ; and the final conclusion would be that the problem did 

 not admit of any solution whatever. How my ingenious 

 friend can for a moment imagine that this example, or indeed 

 that any of the " cluster of instances " contained in Mr. Hor- 

 ner's paper, is either consistent with his view of the matter or 

 inconsistent with mine is to me very extraordinary. For my 

 part I am convinced that any impartial person who gives the 

 slightest attention to the case will be led to an exactly oppo- 

 site conclusion. 



I have before observed that Prof. Young throughout the 

 whole of his letter goes entirely on the wrong hypothesis that 

 analytical ])rocesses under all circumstances must necessarily 

 lead to justifiable residts. I beg again to advance the con- 



