86 Mr DE morgan ON THE GENERAL EQUATION OF 



the new equation are [a], [«], &c.; the functions of these coefficients 

 already noticed are [«J, [/„], [F,], &c. Since the principal diameters 

 of the surface are the same, from whatever equation they are derived, 



w r w'l 



that is, since — 'T ~ ~ rlT ' ^^' *^^ roots of (34) bear to those of [34] 



the proportion of W^ to [ W^ ; whence, \ being an indeterminate quan- 

 tity, since one coefficient in (3) is indeterminate. 



.(39), 





LjO'^'k' M-^' 







These equations* correspond to the general relations (6), (7), and (9), 

 given in my former paper, and from them may be deduced the pro- 

 perties of systems of conjugate diameters, and the remarkable property 

 of the reciprocal squares of three semi-diameters at right angles to one 

 another. 



Let wT', V, and Z', be the co-ordinates of the second origin referred 

 to the first, so that if the co-ordinates be changed, [y] and (p{JC', Y', Z') 

 will be corresponding terms of two equations, the terms of which should 

 be respectively proportional. Assume X, the indeterminate quantity 

 above-mentioned, so that 



[/] = \4>{X', Y', Z') (40). 



and multiply together the first and last of (39), recollecting that 



W 



= -r.-^f^ [^i = -[Fj^t/]. 



* These relations have been given by M. Cacchv, for the case of rectangular co- 

 ordinates, in his " Lcfons sur les applications du Calcnl Infinitesimal d la Geometrie," Vol. i. 

 p. 2441. The equation (34) of this paper, in as general a form, has also been given, since 

 this was written, by Mr Lubbock, in the Philosophical Magazine. 



