Mr. Galbraith on the Figure of the Earth. 163 
principles, it may be supposed that the deviation from the truth 
would be nearly insensible. 
But accurate observations on the length of the pendulum 
at stations selected with judgement, are not of themselves 
the only data necessary to determine the compression. If 
there is any error in the fundamental formula employed for 
this purpose, a corresponding error will be communicated to 
the amount of compression. It is true that the deductions 
hitherto given have the appearance of great accuracy ; for 
when expressed by a fraction of which the numerator is unit, 
the denominator is frequently carried to one or two places 
of decimals. This, however, is a mere deception, if it can 
be shown that the denominator of that fraction is by the or- 
dinary formula erroneous to the amount of several units. 
The formula first demonstrated by Clairaut has been ac- 
quiesced in by Laplace, Delambre, Borda and Biot in France, 
and by Kater and Sabine in England. Now it is well known 
that it is but an approximation obtained in course of the ana- 
lysis by omitting the powers greater than the first.‘ At the 
present time,” says Mr. Ivory (Phil. Mag. vol. lxvi. p. 432), 
one of the ablest geometers of the age, “ when so much has 
been done, and is still doing, to determine the figure of the 
earth experimentally, it seems proper likewise to reconsider 
the theory.” With these sentiments our opinion perfectly 
coincides. The failure of Clairaut’s first attempts to integrate 
a differential equation in the solution of the famous problem 
of the three bodies in his theory of the moon, is well known 
to geometers, and might have suggested the propriety of ex- 
amining this celebrated theorem, and to determine the degree 
of its accuracy by taking in at least another term compre- 
hending the squares in the series expressing the ratio of the 
centrifugal force to gravity. We intended originally to have 
given a complete analysis of this theorem from first principles ; 
but since the time which we are enabled to devote to such 
speculations is but limited, we shall content ourselves by re- 
ferring to a very masterly paper by Mr. Ivory in the Philoso- 
phical Transactions for 1824. It is there demonstrated that 
® sine Qh svi 
q =; sin’? + =, sin* ¢ ik She pleat an Se 
Now when the oblate spheroids do not differ very considera- 
bly from spheres, as in the case of the planets; a, which is 
equal to the eccentricity of the meridian divided by half the 
polar axis is so small that we may consider A* as equal to sin® 
of Silas 2 2 
&c., and in this case =~ al has Miiritsy . . (2) 
. ‘ 25 
Now by reversion of series = 2. Gir a5 (3) 
