Mr. J. Cockle on Impossible Equations* 281 



be stated with reference to that of nitrate of potash, for the 

 selected times, as follows : — 



Nitrate of potash, 1 and 2 per cent, solutions . 100 

 Hydrate of potash, 1 per cent, solution . . . 101*3 

 Hydrate of potash, 2 per cent, solution . . . 99'4< 



These experiments at the low temperature concur, there- 

 fore, with those made at the higher temperature, in proving 

 that the times of equal diffusion of the two substances have 

 been properly chosen. 



[To be continued.] 



XXXI. On Impossible Equations, on Impossible Quantities, and 

 on Tessarines. By James Cockle, Esq., M.A., of Trinity 

 College, Cambridge; Barrister-at~Law,qfthe Middle Temple*. 

 [Continued from vol. xxxvi. p. 292.] 



DEFINITIONS. By an impossible equation is meant an 

 equation which has no root whatever capable of being 

 expressed in terms of the symbols of the ordinary Double 

 Algebra. By an impossible quantity is meant the new species 

 of imaginary by which an impossible equation is supposed to 

 be satisfied. 



* Communicated by T. S. Davies, Esq., F.R.S. Lond. and Ed., who 

 adds the following note. 



"From my having become accidentally involved in the discussion of 

 e congeneric surd equations ' (though merely from having called the atten- 

 tion of Mr. Horner to Garnier's equation, and not from any contribution 

 of my own towards its elucidation), several of my friends, and some gen- 

 tlemen who were strangers, have addressed their views on the subject pri- 

 vately to me. Those of Mr. Cockle, from the somewhat close agreement 

 with my own, and from the form suitable for publication in which they 

 were drawn up, I have sent for insertion in the Philosophical Magazine. 

 Most others were put in forms that would have required modification for 

 the purpose ; and this I did not feel myself at liberty to make, lest I should 

 fail to express in my own language the exact view of the writers. There 

 is one friend, however, a very eminent analyst, who takes a view directly 

 opposed to these; and he has given meat different times his own explana- 

 tion of most of the equations that have been hitherto mooted. When I 

 state that, being opposed to his views (the opposition being founded, as I 

 conceive, on the general principles of analysis), and yet having been uni- 

 formly unsuccessful in detecting any specific fallacy in his reasoning, I can- 

 not view the question as being settled. On this account it is that I think 

 the discussion should be kept open ; and I trust that the gentleman to 

 whom I refer will afford us the benefit of having his * vein of thought' laid 

 open in his own way. Exceptions and failing cases are always the most 

 instructive subjects of inquiry in every science : — in analysis they always 

 betoken something yet left to be seen or done. 



" On one point, however, I wish to be distinctly understood; viz. as not 

 expressing the slightest opinion, at present, on the subject of Quaternions, 

 Tessarines, or any of the inquiries into which i,j Tc enter." 



