On the Equation in Numbers A.r^+ B7/^+ C2^=D.27/2;. 293 



which includes the formula (c), and is now for the first time 

 published. 



This formula (g.) is, however, seen to be a very easy and 

 immediate consequence from the author's fundamental equa- 

 tions of IS-l-a, or from the relations (a.) of the foregoing article, 

 which admit of being concisely summed up in the following 

 continued equation: 



t^=:f = k^ = ijk=-l (h.) 



The geometrical interpretation of the equation S. vt«xt = of 

 the lines of curvature on the ellipsoid, with some other appli- 

 cations of quaternions to that important surface, must be re- 

 served for future articles of the present series, of which some 

 will probably appear in an early number of this Magazine. 

 [To be continued.] 



XLVII. On the Equation in Numbers Aa^ + By^ + Cz^^Dayz, 

 atid its associate si/stem of Equatio7is. By J. J. Sylvester, 

 Esq., M.A., F.R.S* 



[Continued from p. 191.] 



IN the last Number of this Magazine I gave an account of 

 a remarkable transformation to which the equation 



is subject when certain conditions between the coefficients A, 

 B, C, D are satisfied ; which conditions I shall begin by ex- 

 pressing with more generality and precision than I was enabled 

 to do in my former communication. 



1. Two of the quantities A, B, C are to be to one another 

 in the ratio of two cubes. 



2. 27 ABC — D^ must contain no positive prime factor what- 

 ever of the form 6n + }. I erred in my former communication 

 in not excluding cubic factors of this form. 



3. If 2"^ is the highest power of 2 which enters into ABC, 

 and 2" the highest power of 2 which enters into D, then either 

 m must be of the form 3«+ 1, or if not, then m must be greater 

 than 3w. 



These three conditions being satisfied, the given equation 

 can always be transformed into another, 



where A'u^ + WtP + Cxn^ = D'uvwy 



A'B'C = ABC D' = D uvw = a. factor of z. 



The consequence of this is, as stated in my former paper, 

 that wherever A, B, C, D, besides satisfying the conditions 

 above stated, are taken so as likewise to satisfy the condition, — 

 P, of ABC being equal to 23'»±', or 2°, of ABC being equal to 

 23m±ip3u±i^ provided in the first case that ABC is also of the 



* Communicated by the Author. 



