114; Messrs. F. C. Calvert and R. Johnson on the 



z from the above two equations, we may obtain a possible value 

 of %/■ from the resulting equation by methods of approximation. 

 Then substituting this value in one of the equations, we must 

 obtain by the same methods, the corresponding possible value of 

 z, such that z + y^/ — I is found to satisfy the equation X\{^) —^• 

 Tlie solution gives -\-y and —y simultaneously, because, as is 

 known by other reasoning, impossible roots enter equations by 

 pairs. Thus a quadratic factor of X\i^') =0 is found,and the equa- 

 tion may be put under the form Q, ■X'ii^)^^- The equation 

 '^^[a:)=0 may be treated in exactly the same manner, and so on, 

 till a number of quadratic factors be found equal to half the di- 

 mensions of %i(a?) =0. Hence if the dimensions be 2n, we shall 

 have %i(^) = Qi . Q2 • Qa • • • Qn = ^* There cannot be more 

 quadratic factors than these, because if there were, the dimen- 

 sions of their product would exceed the dimensions of %i(a:) j 

 and there cannot be different factors from these, because if 

 any other quadratic factor divided Xi('*')j '*■ must also divide 

 Q, . Q2 . Q3 . . . Q„, which is impossible, because, not being iden- 

 tical with any one of these factors, it is prime to each. Conse- 

 quently the number of the impossible roots is equal to the dimen- 

 sions of ;^i(a.') ; and the whole number of roots, possible and impos- 

 sible, is equal to the dimensions of the given equation /(^)=0. 



The complete solution of the equations (f)^{x, y'^) =0 and 

 ->|rj(r, y^) = 0, might give corresponding values of z and y, one 

 or both of which might be impossible, and yet be such that 

 z-\-y\^ — \ would satisfy the equation ^j(a7)=0. It suffices for 

 the foregoing argument, that there will always be one set at 

 least of corresponding possible values of z and y, which will make 

 z-\-y^ — \ satisfy the same equation. 



A method, practically possible, of finding as many roots of 

 any proposed numerical equation as the equation has dimensions, 

 having been indicated, and the impossibility of finding more 

 having also been shown, it may be concluded generally that 

 every equation has as many roots as dimensions. 



Cambridge Observatory, 

 January 19, 1859. 



XVII. On the Hardness of Metals and Alloys. By P. Crace 

 Calvert, M.R.A. of Turin, F.C.S. i^c. ; and Richard 

 Johnson, F.C.S. ^t* 



THE process at present adopted for determining the compara- 

 tive degree of hardness of bodies, consists in rubbing one 

 body against another, and that which indents or scratches the 



* From the Memoirs of the Literary and Philosophical Society of Man- 

 chester, vol. XV. Session 1857-58. 



