102 
Mr. Ivory on the expansion 
Put u—V i—p . sin ©, z=V 1 — fj, 2 . cos <p ; then 
cos 9 = 
sin 9 —V i — pi,* 
U 
sin <p = - — 
V I -v? 
cos ffl = - 7 2 : 
V i 2 
v I — ■//, 
and by substituting these values and expanding the radical 
quantities wherever they occur, y will be transformed into a 
function of [x, u t z, or of three rectangular co-ordinates. The 
former process has been chosen under the idea, that it exhi- 
bits more clearly the quantities which are in a manner extra- 
neous to the function, and are introduced merely for the 
purpose of making it put on a certain form. 
Having now reduced y to a function of three rectangular co- 
ordinates, the developement in question will be obtained by 
the method of indeterminate coefficients already mentioned. 
Nothing more is necessary than to form the quantities 
Q(°h QW, QW, &c., giving to each the most general expres- 
sion, and leaving the coefficients indeterminate ; then the 
sum of these terms will contain the same combinations of 
V i — . sin (p, V i — jx 1 . cos (p , thaty does ; and by making 
the two expressions coincide, we shall obtain,* 
y = Q (0) + Q w + Q <!) • • • + Q (i) • &c. 
Now let us go back to the original expression ofy. By 
following the process described in the second chapter of the 
third book of the Mecanique Celeste , a similar developement of 
that quantity will be obtained. But it is proved, in the same 
chapter, that a given function cannot be developed two diffe- 
* Mecanique Celeste, Tom. ii. p. 42. 
