Mr. Cockle o?i Equatiojis of the Fifth Degree. 51 



Since these conditions are equations involving three unde- 

 termined quantities x, y and z, they may with propriety be 

 denoted by 



^(.r,3/,2) = 0, (2.) 



and 



^{x,y,z)^Q (3.) 



respectively ; and, since two dimensions of x, y, z appear in 

 (2.), and three in (3.), the former condition will in general 

 represent a surface of the second, and the latter a surface of 

 the third order. 



I have in this work (S. 3. vol. xxviii. pp. 132, 133) already 

 adverted to my own process for reducing equations of the fifth 

 degree to a trinomial form. With what facility such reduc- 

 tion may be performed, at least so far as the decomposition of 

 (2.) into linear factors is concerned, will be seen on referi'ing 

 to my papers on analytical geometry in the last and in the 

 current volume of the Mechanics' Magazine. 



It would not be difficult to show that when (2.) represents 

 a hyperbolic paraboloid, a hyperboloid of one sheet, a cone, 

 a cylinder, a pair of planes, a single plane, or a single straight 

 line, then (2.) and (3.) can be satisfied simultaneously by real 

 values of .r, y, and z. In this case C4 and C- will be real. 



When (2.) represents a hyperboloid of two sheets, an ellip- 

 tic paraboloid, an ellipsoid, a point, or an unreal surface, (2.) 

 and (3.) cannot be so satisfied, and C4 and C5 will be in 

 general both unreal. 



Should C4 and C5, or either of them, prove to be unreal 

 under the above assumption for v, it might be a question 

 whether both those quantities would not take real values under 

 some other assumption, such as 



V = jr?i* + j/?r + zii^ + 1? ; 



but I am very strongly of opinion that they would not, and I 

 am inclined to think that the reality of C4 and Cg depends 

 upon the nature of the roots of the given equation in u. [There 

 is perhaps some analogy between the question before us and 

 that of the irreducible case in cubic equations.] If this be 

 the correct view, no change of process can affect the final re- 

 sult; and we may apply, not only to Mr. Jerrard's, but to the 

 above, and to every possible method of effecting this transfor- 

 mation, the remark of Sir W. R. Hamilton, that the coeffi- 

 cients of the transformed equation will often be unreal when 

 those of the given one are real. 



II. This a]:)pears to be the proper place for explaining an 

 error into which I fell at page 132 of a previous volume of this 

 work (vol. xxviii. just cited). I there omitted to notice that 



E2 



