620 
Hence 
w S . (« + <3 g) (j + <??)(& + <1 £) = V. (j + <1 jj) (A; + <| <£) Si&q + &c. 
From this we may easily verify the former expression by omitting 
products of g, jj, £ 
Thus 
where 
— 1 — A) = jj^i (1 + K) — — Siaq + &c. + &c. ; 
Or 
7 dg cfy dP 
h — H . 
a* ay dz 
,v cftr* 
&c. 
w — — (iSi&q + &c.) + (S . Wji) 
= w 1 + (S . co L <1 )cn as before, . . . (11). ' 
Thus it appears that the equation to the ellipsoid may be written 
T (a + (S« <1 ) <r .) = e (10). 
III. The differential of this equation is 
s(« + (S«<d) <n ^ ^dco + (Sc?w<j ) cr '^ = 0, 
whence, omitting the second order of small quantities, the normal 
vector is 
co + (S« <3 )a~ + <1 Swcn. 
To find the axes we must therefore express that the latter is 
parallel to «, or 
pco = (Sco <| )o~- + <! Swcr- ..... (12). 
where p is an undetermined scalar. 
The most obvious method of solving this equation is to operate 
in succession by S . i, S .j, and S . k. We thus obtain, 
pSico = Sw <1 S?V- 4- Si <1 Scocr- 
&c. = &c. 
Or, remembering (5), 
&c. = 0, 
p is therefore a root of the equation 
