xU 



-y being an arbitrary constant, and u' a variable vector ; and 

 since it must evidently give simpler and more symmetric 

 results to suppose the vector y co-axal with £, than to make 

 the contrary supposition, since we shall thus place the ori- 

 gin of the new vectors a' upon the axis of revolution of the 

 surface, let 



E-y — -yE =r Oj or -y z: ge, (25) 



g being an arbitrary scalar, to be disposed of according to con- 

 venience. Equations (24) and (s?5), cdmbined with (6) and 

 (17), will give, for every point of space, 



_ a'2 = - (a - 7)2 = r^ + g'^e'^ + ^ (aa + ta) ; (26) 

 and therefore, for every point of the locus (15), 



- a'' = r' - 2gr + g''e^ + 2gp. (27) 



The second member of this last equation may be made an 

 exact square, by assuming 



/ e^ + 2gp = g\ that is, g zz j-^ = 2a ; (28) 

 the scalar quotient 



P 



—^ — 2 — ^i ^^ th^ transformation p — a{l — e^), (29) 



being thus suggested to our attention ; and with this value of 

 g we shall have, by (27), 



- a'^ zz (2a - r)^ (30) 



that is, 



2a= v{- c.^)± V(-0; (31) 



so that either the sum or the difference of the distances of any 

 point of the locus (15) from the two foci of which the vectors 

 are respectively and 2a£, is equal to the constant 2a. It is- 

 not difficult to prove that the upper or the lower sign is to be 

 taken, in the formula (31), according as e'^ is < or > 1. For 

 the case e^ = 1, the recent transformation fails. 



Again, to find whether the locus has a centre, we may 

 make 



a = y + 8 =: S''^ + S, (32) 



